Cartesian Products

8/3/2019 Cartesian Products

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Cartesian Products & Binary Relations: An ordered n-tuple (a1, a2, , an) is the ordered collection that has

a1 as its first element, a2 as its second element, and an as its nth element.

2-tuples are called ordered pairs.

(a,b) = (c,d)if and only ifa=c and b=d

(a,b)(b,a)unless a=b

The Cartesian productof two sets A and B denoted by A x B is the set of ordered pairs of the form (a,b), where a is

taken from A and b is taken from B.

This can be written mathematically:

A x B = {(a,b) | a A and b B}

Definition:The Cartesian productof the setsA1, A2, ,An, denotedA1A2 An, is the set of ordered n-

tuples (a1, a2, , an), where aibelongs to set Aifori =1, 2, , n.

A1A2 An={(a1, a2, , an)| aiAi fori=1, 2, , n}Let, A = {0,1,2},B = {a,b}

A B = {(0,a),(0,b),(1,a),(1,b),(2,a),(2,b).

The Cartesian product of sets A and B is denoted AB and defined as follows:

AB ={(x,y): xA and yB}

A={1,3,5,7}, B={2,4,6}

AB={(1,2),(1,4),(1,6),(3,2),(3,4),(3,6), (5,2),(5,4),(5,6),(7,2),(7,4),(7,6)}

Binary Relations: Let A and B be sets. A binary relation between two sets A and B is a SUBSET of A x B.

A binary relation from A to B is a set R of ordered pairs where the first element of each ordered pair comes from A

and the second element comes from B. A relation, in general, is any set of ordered n-tuples chosen from n sets.

If(a,b) R, then we say a is related to b by R. This is sometimes written as a R b. Definition: LetA andB be sets. A binary relation R from A to B is a subset of the Cartesian productA B. Notation:xRy means (x,y)R, and

xis said to be related toy underR.

xRy denotes (x,y)R.Example:

Examples Let A = {1, 2, 3} and B = {a, b}

Relation from A to B?

R = {(1, 1) (2, 2)} No

R = { } Yes


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R = {(1, a) (1, b)} Yes

R = {(a, 1) (b, 2)} No

Let A = {1, 2, 3}

R = { } Yes

R = {(1, 2) (2, 3) (3, 4) No

Relations on a set: A relation on the set A is a relation from A to A. A relation on a set is a subset of A x A.

Inverse Relations: Given a relation RAB then R-1BAsuch that:R-1={(y,x):(x,y)R}R= {(x,y):x


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Domain of R: Domain of a Relation(R) is the set of all the first elements of ordered

pairs of the Relation(R). Let R be a subset of A A, we define domain of R: DR = {a

A(a; b) R}.

Range of R: Range of a Relation(R) is the set of all the second elements of ordered

pairs of The Relation(R). Let R be a subset of A A, we define range of R: RR = { b A(a;b) R}.

For Example:

If a Relation R=[{1,2} ,{2,3} ,{3,4}]

Domain of Relation R ={1,2,3}

And Range of Relation R={2,3,4}

n-ary: LetA1,A2, ,An be sets. An n-ary relation on these sets (in this order) is a subset ofA1A2 An.

Ordered tuple :Given setsA1; :::An. An element (a1; a2; :::an) such that ai Ai for i = 1; 2; :::nis calledan ordered TUPLE.

Example:

IfX = {1, 2, 3, 4, 5, 6} , findR = { (x,y): x is a divisor of y}

R = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,2), (2,4), (2,6), (3,3), (3,6) }

Arrow Diagrams of Relations: Example 1

Draw an arrow diagram from x to yxRy(x, y) R

Let A = {-1, 0, 1} and B = {1, 2, 3} and xRyxy 1

R = {(1, 1) (1, 2) (1, 3)}


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Example 2

Draw an arrow diagram from x to y xRy (x, y) R Let A = {1, 2, 3, 4} and B = {1, 2} and xRy x = y2 R = {(1, 1) (4, 2)}

Not a function because every element in A is not the first element of an ordered pairNote: Relations can have

elements in A with no arrow coming out of it and an element in A with multiple arrows pointing to different

elements in B *this is different than a function*

Directed Graph of Relations:

Remember: A binary relation on a set A is a binary relation from A to A

The graph of this relation is called a directed graph

Example

Let A = {1, 2, 3} (x, y)A, xRy x y A x A = {(1, 1) (1, 2) (1, 3) (2, 1) (2, 2) (2, 3) (3, 1) (3, 2) (3, 3)}

R = {(1, 1) (2, 1) (2, 2) (3, 1) (3, 2) (3, 3)}.

Directed Graph of Relations (contd)

Draw the directed graph of the binary relation


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Let A = {2, 3, 4, 5, 6, 7, 8} (x, y)A, xTy 3 | (x-y)A x A = { (2,2) (2,3) (2,4) (2,5) (2,6) (2,7) (2,8), (3,2) (3,3) (3,4) (3,5) (3,6) (3,7) (3,8), (4,2) (4,3) (4,4) (4,5) (4,6)

(4,7) (4,8), (5,2) (5,3) (5,4) (5,5) (5,6) (5,7) (5,8), (6,2) (6,3) (6,4) (6,5) (6,6) (6,7) (6,8), (7,2) (7,3) (7,4) (7,5) (7,6)

(7,7) (7,8), (8,2) (8,3) (8,4) (8,5) (8,6) (8,7) (8,8) }

Matrix representation of relation: If A and B are finite sets and R is a binary relation between A and B then

create a matrix, M, with the following properties:

the rows of the matrix are indexed by the elements of A

the columns of the matrix are indexed by elements of B

M(ai,bj) = 1 if (ai,bj) belongs to R; 0 otherwise

R = {(a,b): a + 1 < b}

Consider the set S= {1, 2, 3, 4, 5, 6, 7, 8}.

The relation < is the set of ordered pairs

R = {(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), (2, 8), (3, 4),

(3, 5), (3, 6), (3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8), (5, 6), (5, 7), (5, 8), (6, 7), (6, 8), (7, 8)}.

< 1 2 3 4 5 6 7 8

1 2 3 4


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5 6 7 8

Graph Representation of a Binary Relation (Directed graph): IfA andB are two finite sets andR is a binary

relation betweenA andB we can represent this relation as a graph (set of vertices and edges).

A = {1, 3, 5 ,7}, B = {2, 4, 6, 8} . R = {(a,b): a < b}

Properties on Relations:

Reflexive

irreflexive

Symmetric

Antisymmetric

Transitive

Asymmetric

Reflexive: A relation R on a set A is called reflexive if (a,a) R forevery element a A.

Irreflexive: R is irreflexive means: (x, x) oe R for all x belongs to A.

Symmetric: A relation R on a set A is called symmetric if (b,a) R whenever (a,b) R, for some a,b A.

Antisymmetric: A relation R on a set A such that (a,b) R and (b,a) R only if a = b for a,b A is called

antisymmetric. Note that antisymmetric is not the opposite of symmetric. A relation can be both.

Asymmetric: A relation R on a set A is called asymmetric if (a,b) R (b,a) R.


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Transitive: A relation R on a set A, is called transitive if whenever (a,b) R and (b,c) R, then (a,c) R , for a, b,

c A.

Consider the following relations on {1,2,3,4}. Which are reflexive, irreflexive, neither?

R1 = {(1,1), (1,2), (2,1), (2,2), (3,3), (4,4)}

Reflexive R2 = {(2,4), (4,2)}

Irreflexive

R3 = {(1,2), (2,3), (3,4)}

Irreflexive

R4 = {(1,1), (2,2), (3,3), (4,4)}

Reflexive

R5 = {(1,3), (1,4), (2,3), (2,4), (3,1), (3,4)}

Irreflexive

R6 = {(2,2), (2,3), (2,4), (3,2), (3,3), (3,4)} Neither

Consider the following relations on {1,2,3,4}. Which are transitive? R1 = {(2,2), (2,3), (2,4), (3,2), (3,3), (3,4)}

Transitive

R2 = {(1,1), (1,2), (2,1), (2,2), (3,3), (4,4)}

Transitive

R3 = {(2,4), (4,2)}

Not Transitive

R4 = {(1,2), (2,3), (3,4)}

Not Transitive

R5 = {(1,1), (2,2), (3,3), (4,4)}

Transitive

R6 = {(1,3), (1,4), (2,3), (2,4), (3,1), (3,4)}

Not Transitive

The divides relation on the set of positive integers

Transitive

Consider the following relations on {1,2,3,4}. Name its properties?

R1 = {(2,2), (2,3), (2,4), (3,2), (3,3), (3,4)}

T/A

R2 = {(1,1), (1,2), (2,1), (2,2), (3,3), (4,4)}

R/S/T R3 = {(2,4), (4,2)}

S

R4 = {(1,2), (2,3), (3,4)}

A

R5 = {(1,1), (2,2), (3,3), (4,4)}

R/S/A/T


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R6 = {(1,3), (1,4), (2,3), (2,4), (3,1), (3,4)}

None

The divides relation on the set of positive integers

R/A/T

Keys: R = Reflexive, S = Symmetric, A = Antisymmetric, T=Transitive

Determine the properties of each of the following relations defined on the set of all real numbersR:

R = {(x,y) |x +y = 0}

R = {(x,y) |x =y x = y}

R = {(x,y) |x y is a rational number}

R = {(x,y) |x = 2y}

R = {(x,y) |xy 0}

R = {(x,y) |xy = 0}

R = {(x,y) |x = 1}

R = {(x,y) |x = 1 ory = 1}

R = {(x,y) |x +y = 0}

Solution:

It is not reflexive since, for example, (1,1) R.

It is not irreflexive since, for example, (0,0) R.

Sincex +y =y +x, it follows that ifx +y = 0 theny +x = 0, so the relation is symmetric.

It is not antisymmetric since, for example, (1,1) and

(1, 1) are both inR, but 1 1

The relation is not transitive since, for example,

(1,1) R and (1,1) R, but (1,1) R.

R = {(x,y) |x =y x = y}

Solution:

Since for eachx,xRx then it is reflexive.

Since it is reflexive, it is not irreflexive.

Sincex = y if and only ify = x, then it is symmetric.

It is not antisymmetric since, for example, (1,1)and (1,1) are both inRbut 1 1

It is also transitive because essentially the product of 1 and 1 is 1.

R = {(x,y) |x y is a rational number}

Solution:


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It is reflexive since, for allx, x x = 0 is a rational number.


It is symmetric because, ifx y isrational, then (x y) = y x is also.

It is not antisymmetric because, for example, (1, 1) and ( 1,1) are both inR but 1 1.

It is transitive because ifx y is a rational andy z is a rational, so isx y +y z = x z.

R = {(x,y) |x = 2y}

Solution:

It is not reflexive since, for example, (1,1) R.

It is not irreflexive since, for example, (0,0) R.

It is not symmetric since, for example, (2,1) R

but (1,2) R.

It is antisymmetric becausex =y =0 is the only time that (x,y) and (y,x) are both inR.

It is not transitive since, for example, (4,2) R and (2,1) R but (4,1) R.

R = {(x,y) |x y 0}

Solution:

It is reflexive sincexx 0.


It is symmetric as the role ofx andy are interchangeable.

It is not antisymmetric since, for example, (2,3) and (3,2) are both inR,but2 3.

It is not transitive because, for example, (1,0) and(0, 2) are both inR but (1, 2) is not.

R = {(x,y) |xy = 0}

Solution:

It is not reflexive since (1,1) R.

It is not irreflexive since (0,0) R.

It is symmetric as the role ofx andy are interchangeable.

It is not antisymmetric since, for example, (2,0) and (0,2) are both inRbut2 0.

It is not transitive because, for example, (1,0) and (0, 2) are both inR but (1, 2) is not.

R = {(x,y) |x = 1}

Solution:

It is not reflexive since (2,2) R.

It is not irreflexive since (1,1) R.

It is not symmetric since, for example, (1,2) R but (2,1) R.

It is antisymmetric because if (x,y) R and (y,x) R it meansx =y = 1.

It is transitive since if (1,y) R and (y,z) R, so is (1,z) R .


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Matrix Representation of Binary Relations

Matrices are used to represent the ordered pairs in a Cartesian product, A x B. The rows are

labelled with elements of A and the columns are labeled with elements of B. Each position in the

matrix is referred to by the ordered pair (row label (in A), column label (in B)).

Then, we use this matrix to represent different relations on A x B. When the ordered pair (a,b) is

in the relation R, we place a 1 in position (a,b) in the matrix. When (a,b) is not in R, we place a 0in position (a,b).


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For example:A = {a1,a2,a3,a4}

B = {b1,b2,b3,b4}

R = { (a1,b1), (a2, b1),

(a3,b2), (a3, b3), (a4,b4) }

R in A x B

b1 b2 b3 b4a1 1 0 0 0

a2 1 0 0 0

a3 0 1 1 0

a4 0 0 0 1

Composite Relations

Consider the following problem: You would like to fly from an airport in NC to somewhere near

or in Ireland for a vacation, and you don't care where you fly through, so long as you get betweenNC and Ireland. But there are no direct flights available, so you'll have to take a two-leg trip.

You really don't want to have more legs than that. Determine if there is a way to fly between here

and there.

First, we figure out which cities have available flights leaving from airports in NC, that also have

flights to London or Ireland:

A: Destinations from NC

Atlanta Washington New York


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Greensboro 0 1 0

Raleigh 1 0 0

Charlotte 1 1 1Next, we determine the available flights from Atlanta, Washington, and New York, to cities

near/in Ireland.

B: Destinations in/near Ireland

London Dublin Shannon

Atlanta 0 1 0Washington 1 1 0

New York 1 0 1

Now, to find flights from NC to Ireland, we perform matrix composition (this is matrix

multiplication, replacing multiplies with logical AND and replacing addition with logical OR),

on the two matrices, to get a matrix of values for available trips between Greensboro, Raleigh,

and Charlotte, and London, Shannon, and Dublin.

A o B: NC to in/near Ireland

London Dublin Shannon

Greensboro 1 1 0

Raleigh 0 1 0

Charlotte 1 1 1


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Reflexive and Irreflexive Relations

A binary relation R from the set A to itself is said to be reflexive if, for all elements a in A, (a,a)

is in R.

Example 1: x


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Example 1: x < y is asymmetric

Example 2: "x has more course hours than y" is asymmetric

When can relations be both symmetric and antisymmetric?

How about both symmetric and asymmetric? Are there relations that are none of these? Why?

What do the matrix representations of these relations look like?

Transitive Relations

R in A x A is transitive if, for all a, b, c, in A,

IF (a,b) is in R, AND (b,c) is in R, THEN (a,c) is in R.

Example 1: x < y is transitive. Example 2: "x is taller than y" is transitive. Example 3: x = 2y is NOT transitive.

To determine whether a relation is transitive is much harder than

determining whether a relation is reflexive, irreflexive, symmetric, etc. Remember composite relations? To do a composite of two relations S in A

x B and T in B x C, we get a point (a,c) in (S o T) whenever there is a point (a,k)

in S and a point (k,c) in T, where k is some (random) element in B. To say this adifferent way, if (a,k) is in S, and (k,c) is in T, then (a,c) is in S o T. This should

look very similar to our definition of transitivity.

We can use relation composition to determine if R is transitive. Compose

R o R to get "R squared". This is a new relation. Whenever (a,b) is in R, and (b,c)is in R, then (a,c) is in R o R. So, all the points that should be in R, if R is

transitive, are automatically found in R o R. So, we can tell if R is transitive by

seeing if R o R is a subset of R (i.e. every point in R o R is also a point in R).

Equivalence Relations & Posets

R in A x A is an equivalence relation if R is:

reflexive,

symmetric, and transitive.

Example 1: [x = y (mod 3)] is an equivalence relation. This partitions integers into 3 kinds of

sets: those with remainders of 0 when we divide by 3, those with remainders of 1, and those withremainders of 2. Under this relation, we say 0 is equivalent to 3, 6, 9, ...; 1 is equivalent to 4, 7,

10, ...; and 2 is equivalent to 5, 8, 11, ...


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Example 2: "x and y are in the same discrete math class" is also an ER.

R in A x A is anposet, orpartially ordered set, if R is:

reflexive,

antisymmetric, and transitive.

Example 1: x


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A x B = { ( a , 1 ) , ( a , 2 ) , ( b , 1 ) , ( b , 2 ) }

B x A = { ( 1 , a ) , ( 1 , b ) , ( 2 , a ) , ( 2 , b ) }

Hence A x B B x A

There is one exception with .

A x = x A

For two equal sets P and Q,

P x Q = Q x P , where P = Q

For example, P = { 1 , 2 } and Q = { 1 , 2 }

P x Q = { ( 1 , 1 ) , ( 1 , 2 ) , ( 2 , 1 ) , ( 2 , 2 ) }

Q x P ={ ( 1 , 1 ) , ( 1 , 2 ) , ( 2 , 1 ) , ( 2 , 2 ) }

3. Cartesian product is not associative. For any three non equal sets A , B and C

( A x B ) x C A x ( B x C)

For example , A = { a , b} , B = { c , d }and C ={ e , f } then

( A x B ) x C = { ( ( a , c ) ,e ) , ( ( a , d ) , e ) , ( ( a , c ) , f ) , ( ( a , d ) , f ) , ( ( b , c ) ,e ) , ( ( b , d ) , e ) , ( ( b , c ) , f ) , ( ( b , d ) , f ) }

A x ( B x C ) = { ( a , ( c , e ) ) , ( a , ( d , e ) ) , ( a , ( c , f ) ) , ( a , ( d , f ) ) , ( b , ( c , e) ) , ( b , ( d , e ) )v , ( b , ( c , f ) ) , ( b , ( d , f ) ) }

Hence ( A x B ) x C A x ( B x C)

4. Intersection holds on Cartesian product.

( A B ) x ( C D ) = ( A x C ) ( B x D )

5. The above case in not true for union

( A U B ) x ( C U D ) ( A x B ) U ( C x D )

6. ( A ) x ( B C ) = ( A x B ) ( A x C )

7. ( A ) x ( B U C ) = ( A x B ) U ( A x C )


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n - Ary product

Generalized form of Cartesian product for n number of sets is called n-ary product.X1 , X2 Xn be n sets. Cartesian product of these n sets can be given as,

X1 x X2 x .......Xn = { (x1 , x2...xn ) : xi Xi }

Its a n-tuple set. All tuples are defined by nested ordered pairs.

Cartesian square

Cartesian square is also known as binary Cartesian product. For a set A the Cartesian square is

A2 =A x A. For example a Cartesian plane is a Cartesian square represented by R2 that is R x R

where R is set of real numbers.

For example

set G={ 1 , 2 }

G2 ={ ( 1 , 1 , ) , ( 1 , 2 ) , ( 2 , 1 ) , ( 2 , 2 ) }

Cartesian power

It is a generalized form. Power is represented as a variable n. For a set X and power n,

Xn= X x X x X ....x X (n times) = { ( x1,x2,...xn): x1X x2 X....xn X }

For example , for a set G,

G4 = G x G x G x G

Examples based on Cartesian product

1. Cartesian product of two set A x B has 6 elements If three of these are ( a , b ) , ( b , c ) and

( c , c ) , then find Cartesian product set " B x A ".

Solution:

First elements of ordered pairs of A x B are elements of set A. Hence a , b and c are

the elements of set A, whereas second elements of ordered pairs of A x B are elements of set B.Hence, b and c are elements of set B . It is given that there are total 6 elements in the

Cartesian product. We have 3 elements in set A and 2 in set B , There product is 6 ( i.e., 3 x 2 = 6

). Hence we have got all the elements of set A and set B.

A = { a , b , c }


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B = { b , c }

We have set A and set B , now we can find B x A .

B x A = { ( b , a ) , ( b ,b ) , ( b , c) , ( c , a ) , ( c, b ) , ( c , c ) }

2. Cartesian product of two sets X x Y has 4 elements .If two of these are ( 1 , 2 ) and ( 2 ,

3 ).Find Cartesian product of Y x X.

Solution :

First element of ordered pair of X x Y are elements of set X and second elements of ordered

pair of set X x Y belongs to set Y. Hence 1 and 2 belongs to set X and 2 and 3 are

elements of set Y. So far set X and set Y have 2 elements each. And 2 x 2 is 4. It is given thatCartesian product of set X x Y have 4 elements .It implies that we identified all the elements of

set X and set Y.

We have,

X = { 1 , 2 }

Y = { 2 , 3 }

We have set X and set Y .hence,

Y x X = { ( 2 , 1 ) , ( 2 , 2 ) , ( 3 , 1 ) , ( 3 , 2 ) }

3.Two sets are given as : A = { a , b } and B = { c , d }. Find the total numbers of subsets of A xB and hence write the power set of AXB .

Solution:

Let x be he total number of elements in set A.

x = n ( A ) = 2

Similarly let y be the number of elements of set B.

y = n ( B ) = 2

Now, the total number of elements in Cartesian product of A x B is xy.

n( A x B) = xy = 2 x 2 = 4

A x B = { ( a , c ) , ( a , d ) , ( b , c ) , ( b , d ) }


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Power set of A x B contains 2xy elements. Hence power set of A x B contains 24 i.e., 16 elements.

n( P ( A x B ) ) = 16

P ( A x B ) = {, ( a , c ) , ( a , d ) , ( b , c ) , ( b , d ) , { ( a , c ) , ( a , d ) } , { ( a , c ) , ( b , c ) } ,

{ ( a , c ) , ( b , d ) } , { ( a , d ) , ( b , c ) } , { ( a , d ) , ( b , d ) } , { ( b , c ) , ( b , d ) } , { ( a , c ), ( a , d ) , ( b , c ) } , { ( a , c ) , ( a , d ) , ( b , d) } , { ( a, c ) , ( b, c ) , ( b , d ) } , { ( a , d ) , ( b ,

c ) , ( b , d ) } ,

{ ( a , c ) , ( a , d ) , ( b, c ) , ( b , d ) } }

Related Tags

Explain Cartesian Product Of Two Sets , Introduction to Cartesian Product Of Two Sets , What

is Cartesian Product Of Two Sets

Cartesian Product:

The Cartesian Product of two setsA and B is the set of all Ordered Pairs (a,b) where the first

element of order pairs a belongs to first set A and second element of ordered pairs b

belongs or second set B.

Or aA and bB

Note: Cartesian product of set A and B is not equal to Cartesian product of set B and A.

Denotation of Cartesian product:

Cartesian product of sets A and B is denoted by :

AB

And Cartesian product of sets B and A is denoted by:

BA

For example:

If set A={1,2} and set B={4,5}

Then,

AB=[ {1,4} , {1,5} , {2,4} , {2,5} ]

And
http://oscience.info/mathematics/introduction-to-set/http://oscience.info/mathematics/introduction-to-set/http://oscience.info/mathematics/ordered-pairs/http://oscience.info/mathematics/introduction-to-set/http://oscience.info/mathematics/ordered-pairs/


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BA=[ {4,1} , {4,2} , {5,1} , {5,2} ]

Note: If m is the number of elements in set A and n is number of elements in set B then

the numbers of elements of AB and BA is mn

For example:

If set A have 2 elements and Set B have 3 elements the the number of elements that AB andBA have is 32=6.

What is Relation?

Any subset of a Cartesian product AB in which the first element and second element ofordered pairs have special relation to each other is known as Relation.

A relation from one set (A) to another set (B) is denoted by:

xRy or simply R

Where (x,y)R

For example:

If set A={Me , My father , My son}

And set B={My spouse , My mother , My daughter}

Then one of the Relation from set A to set B can be:

R=[{My spouse , Me} , {My mother , My father}]

In above Relation , the relation between the first and second element of ordered pairs is that first

element is wife of Second element. Like My spouse Is wife of Me

And

If set A={2,3,4}

And set B={4,5,6}

Then one of the Relation from set A to set B can be :

R=[{2,4} , {2,6} ,{3,6} ,{4,4}]

In above Relation the relation between first and second element of ordered pairs is that First

element is a factor of second element. Like 4 is a factor of 4.
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Domain and Range of a Relation:

Domain of a Relation(R) is the set of all the first elements of ordered pairs of the Relation(R)

and

Range of a Relation(R) is the set of all the second elements of ordered pairs of The

Relation(R).

For Example:

If a Relation R=[{1,2} ,{2,3} ,{3,4}]

Domain of Relation R ={1,2,3}

And Range of Relation R={2,3,4}

Diagrammatically we can denote the relations from one set(A) to another (B) as following:

,


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etc.

Inverse Relations:

A relation obtained by interchanging first and second elements in the ordered pairs of given

Relation is known as the inverse Relation of given Relation.

If a Relation R is given then the inverse of the relation R is denoted by the symbol:

R-1

For example:

If Relation R=[{1,2} , {3,4} , {5,6}}

Then the inverse of Relation R =R-1=[{2,1} , {4,3} , {6,5}]

Relation (mathematics) redirects here. For a more general notion of relation, see Finitary

relation. For a more combinatorial viewpoint, see Theory of relations.

In mathematics, a binary relation on a setA is a collection ofordered pairs of elements ofA. In

other words, it is a subset of the Cartesian productA2 =A A. More generally, a binary relation

between two setsA andB is a subset ofA B. The terms dyadic relation and 2-place relationare synonyms for binary relations.

An example is the "divides" relation between the set ofprime numbersP and the set ofintegers

Z, in which every primep is associated with every integerzthat is a multiple ofp (and not with

any integer that is not a multiple ofp). In this relation, for instance, the prime 2 is associated with

numbers that include 4, 0, 6, 10, but not 1 or 9; and the prime 3 is associated with numbers that

include 0, 6, and 9, but not 4 or 13.
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Binary relations are used in many branches of mathematics to model concepts like "is greater

than", "is equal to", and "divides" in arithmetic, "is congruent to" in geometry, "is adjacent to" in

graph theory, "is orthogonal to" inlinear algebra and many more. The concept offunction isdefined as a special kind of binary relation. Binary relations are also heavily used in computer

science.

A binary relation is the special case n = 2 of an n-ary relationRA1 An, that is, a set ofn-tuples where thejth component of each n-tuple is taken from thejth domainAj of the relation.

In some systems ofaxiomatic set theory, relations are extended to classes, which aregeneralizations of sets. This extension is needed for, among other things, modeling the concepts

of "is an element of" or "is a subset of" inset theory, without running into logical inconsistencies

such as Russell's paradox.

Formal definition

A binary relationR is usually defined as an ordered triple (X, Y, G) whereXand Yare arbitrarysets (or classes), and G is a subset of the Cartesian productX Y. The setsXand Yare called the

domain (or the set of departure) and codomain(or the set of destination), respectively, of the

relation, and G is called its graph.

The statement (x,y) R is read "xisR-related toy", and is denoted byxRy orR(x,y). The latternotation corresponds to viewingR as the characteristic functionon "X" x "Y" for the set of pairs

ofG.

The order of the elements in each pair ofG is important: ifa b, then aRb and bRa can be trueor false, independently of each other.

Is a relation more than its graph?

According to the definition above, two relations with the same graph may be different, if theydiffer in the setsXand Y. For example, ifG = {(1,2),(1,3),(2,7)}, then (Z,Z, G), (R, N, G), and

(N, R, G) are three distinct relations.

Some mathematicians do not consider the setsXand Yto be part of the relation, and therefore

define a binary relation as being a subset ofXY, that is, just the graph G. According to thisview, the set of pairs {(1,2),(1,3),(2,7)} is a relation from any set that contains {1,2} to any set

that contains {2,3,7}.

A special case of this difference in points of view applies to the notion of function. Most authors

insist on distinguishing between a function'scodomain and itsrange. Thus, a single "rule" likemapping every real numberx tox2 can lead to distinct functionsf:RR andg:RR+, depending

on the images under that rule are understood to be reals or, more particularly, non-negative reals.

But others view functions as simply sets of ordered pairs with unique first components. Thisdifference in perspectives does raise some nontrivial issues. As an example, the former camp
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considers surjectivityor being ontoas a property of functions, while the latter sees it as a

relationship that functions may bear to sets.

Either approach is adequate for most uses, provided that one attends to the necessary changes inlanguage, notation, and the definitions of concepts like restrictions,composition,inverse relation,

and so on. The choice between the two definitions usually matters only in very formal contexts,like category theory.

Example

Example: Suppose there are four objects {ball, car, doll, gun} and four persons {John, Mary, Ian,

Venus}. Suppose that John owns the ball, Mary owns the doll, and Venus owns the car. Nobody

owns the gun and Ian owns nothing. Then the binary relation "is owned by" is given as

R=({ball, car, doll, gun}, {John, Mary, Ian, Venus}, {(ball, John), (doll, Mary),

(car, Venus)}).

Thus the first element of R is the set of objects, the second is the set of people, and the last

element is a set of ordered pairs of the form (object, owner).

The pair (ball, John), denoted by ballRJohn means that the ball is owned by John.

Two different relations could have the same graph. For example: the relation

({ball, car, doll, gun}, {John, Mary, Venus}, {(ball,John), (doll, Mary), (car,

Venus)})

is different from the previous one as everyone is an owner. But the graphs of the two relationsare the same.

Nevertheless,R is usually identified or even defined as G(R) and "an ordered pair (x,y) G(R)"is usually denoted as "(x,y) R".

Special types of binary relations

Some important classes of binary relationsR betweenXand Yare listed below.

Uniqueness properties:

injective (also called left-unique[1]): for allxandzinXandyin Yit holdsthat ifxRyandzRythenx=z.

functional (also called right-unique[1] or right-definite[citation needed]): for allxinX, andyandzin Yit holds that ifxRyandxRztheny=z; such a binaryrelation is called a partial function.

one-to-one (also written 1-to-1): injective and functional.
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Totality properties:

left-total[1]: for allxinXthere exists ayin Ysuch thatxRy(this property,although sometimes also referred to as total, is different from the definitionoftotal in the next section).

surjective (also called right-total[1]): for allyin Ythere exists anxinXsuchthatxRy.

A correspondence: a binary relation that is both left-total and surjective.

Uniqueness and totality properties:

A function: a relation that is functional and left-total. A bijection: a one-to-one correspondence; such a relation is a function and is

said to be bijective.

Relations over a set

IfX= Ythen we simply say that the binary relation is overX. Or it is an endorelation overX.Some classes of endorelations are widely studied ingraph theory, where they're known as

directed graphs.

The set of all binary relationsB(X) on a set X is a semigroup with involution with the involutionbeing the mapping of a relation to its inverse relation.

Some important classes of binary relations over a setXare:

reflexive: for allxinXit holds thatxRx. For example, "greater than or equalto" is a reflexive relation but "greater than" is not.

irreflexive (or strict): for allxinXit holds that notxRx. "Greater than" is anexample of an irreflexive relation.

coreflexive: for allxandyinXit holds that ifxRythenx=y. "Equal to andodd" is an example of a coreflexive relation.

symmetric: for allxandyinXit holds that ifxRythenyRx. "Is a bloodrelative of" is a symmetric relation, becausexis a blood relative ofyif andonly ifyis a blood relative ofx.

antisymmetric: for allxandyinXit holds that ifxRyandyRxthenx=y."Greater than or equal to" is an antisymmetric relation, because ifxyand

yx, thenx=y. asymmetric: for allxandyinXit holds that ifxRythen notyRx. "Greater

than" is an asymmetric relation, because ifx>ythen noty>x. transitive: for allx,yandzinXit holds that ifxRyandyRzthenxRz. "Is an

ancestor of" is a transitive relation, because ifxis an ancestor ofyandyisan ancestor ofz, thenxis an ancestor ofz.

total: for allxandyinXit holds thatxRyoryRx(or both). "Is greater than orequal to" is an example of a total relation (this definition for total is differentfrom the one in the previous section).

trichotomous: for allxandyinXexactly one ofxRy,yRxorx=yholds. "Isgreater than" is an example of a trichotomous relation.
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IfR is a binary relation overXand Y, and Sis a binary relation overYandZ, then the following

is a binary relation overXandZ: (see main article composition of relations)

Composition: S R, also denoted R;S (or more ambiguously R S), definedas S R = { (x,z) | there existsy Y, such that (x,y) R and (y,z) S }.

The order ofR and S in the notation S R, used here agrees with the standardnotational order for composition of functions.

Complement

IfR is a binary relation overXand Y, then the following too:

The complementS is defined asxSyif notxRy.

The complement of the inverse is the inverse of the complement.

IfX= Ythe complement has the following properties:

If a relation is symmetric, the complement is too. The complement of a reflexive relation is irreflexive and vice versa. The complement of a strict weak order is a total preorder and vice versa.

The complement of the inverse has these same properties.

Restriction

The restrictionof a binary relation on a setXto a subset Sis the set of all pairs (x,y) in the

relation for whichx andy are in S.

If a relation is reflexive, irreflexive, symmetric,antisymmetric, asymmetric, transitive,total,trichotomous, apartial order, total order,strict weak order, total preorder(weak order), or an

equivalence relation, its restrictions are too.

However, the transitive closure of a restriction is a subset of the restriction of the transitive

closure, i.e., in general not equal.

Also, the various concepts ofcompleteness(not to be confused with being "total") do not carry

over to restrictions. For example, on the set ofreal numbers a property of the relation "" is that

every non-empty subset SofRwith an upper boundin Rhas aleast upper bound (also called

supremum) in R. However, for a set of rational numbers this supremum is not necessarilyrational, so the same property does not hold on the restriction of the relation "" to the set of

rational numbers.

The left-restriction (right-restriction, respectively) of a binary relation betweenXand Yto asubset Sof its domain (codomain) is the set of all pairs (x,y) in the relation for whichx (y) is an

element ofS.
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Sets versus classes

Certain mathematical "relations", such as "equal to", "member of", and "subset of", cannot beunderstood to be binary relations as defined above, because their domains and codomains cannot

be taken to be sets in the usual systems ofaxiomatic set theory.

For example, if we try to model the general concept of "equality" as a binary relation =, we must

take the domain and codomain to be the "set of all sets", which is not a set in the usual set theory.The usual work-around to this problem is to select a "large enough" set A, that contains all the

objects of interest, and work with the restriction =A instead of =.

Similarly, the "subset of" relation needs to be restricted to have domain and codomainP(A)(the power set of a specific setA): the resulting set relation can be denoted A. Also, the"member of" relation needs to be restricted to have domainA and codomainP(A) to obtain a

binary relation A that is a set.

Another solution to this problem is to use a set theory with proper classes, such asNBG orMorseKelley set theory, and allow the domain and codomain (and so the graph) to beproper

classes: in such a theory, equality, membership, and subset are binary relations without special

comment. (A minor modification needs to be made to the concept of the ordered triple (X, Y, G),as normally a proper class cannot be a member of an ordered tuple; or of course one can identify

the function with its graph in this context.)

In most mathematical contexts, references to the relations of equality, membership and subset are

harmless because they can be understood implicitly to be restricted to some set in the context.

The number of binary relations

The number of distinct binary relations on an n-element set is 2n2 (sequence A002416 in OEIS):

Number ofn-element binary relations of different types

n alltransit

ive

reflexi

ve

preord

er

partial

order

total

preorder

total

order

equivalence

relation

0 1 1 1 1 1 1 1 1

1 2 2 1 1 1 1 1 1

2 16 13 4 4 3 3 2 2

3 512 171 64 29 19 13 6 5

4 65536 3994 4096 355 219 75 24 15

OEI

S

A0024

16

A00690

5

A0537

63

A0007

98A001035 A000670 A000142 A000110
http://www.enotes.com/topic/Axiomatic_set_theoryhttp://www.enotes.com/topic/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theoryhttp://www.enotes.com/topic/Morse%E2%80%93Kelley_set_theoryhttp://www.enotes.com/topic/Proper_classhttp://www.enotes.com/topic/Proper_classhttp://www.enotes.com/topic/Proper_classhttp://www.research.att.com/~njas/sequences/A002416http://www.enotes.com/topic/On-Line_Encyclopedia_of_Integer_Sequenceshttp://www.enotes.com/topic/Transitive_relationhttp://www.enotes.com/topic/Transitive_relationhttp://www.enotes.com/topic/Reflexive_relationhttp://www.enotes.com/topic/Reflexive_relationhttp://www.enotes.com/topic/Preorderhttp://www.enotes.com/topic/Preorderhttp://www.enotes.com/topic/Partially_ordered_sethttp://www.enotes.com/topic/Partially_ordered_sethttp://www.enotes.com/topic/Weak_orderhttp://www.enotes.com/topic/Weak_orderhttp://www.enotes.com/topic/Total_orderhttp://www.enotes.com/topic/Total_orderhttp://www.enotes.com/topic/Equivalence_relationhttp://www.enotes.com/topic/Equivalence_relationhttp://www.enotes.com/topic/OEIShttp://www.enotes.com/topic/OEIShttp://www.research.att.com/~njas/sequences/A002416http://www.research.att.com/~njas/sequences/A002416http://www.research.att.com/~njas/sequences/A006905http://www.research.att.com/~njas/sequences/A006905http://www.research.att.com/~njas/sequences/A053763http://www.research.att.com/~njas/sequences/A053763http://www.research.att.com/~njas/sequences/A000798http://www.research.att.com/~njas/sequences/A000798http://www.research.att.com/~njas/sequences/A001035http://www.research.att.com/~njas/sequences/A000670http://www.research.att.com/~njas/sequences/A000142http://www.research.att.com/~njas/sequences/A000110http://www.enotes.com/topic/Axiomatic_set_theoryhttp://www.enotes.com/topic/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theoryhttp://www.enotes.com/topic/Morse%E2%80%93Kelley_set_theoryhttp://www.enotes.com/topic/Proper_classhttp://www.enotes.com/topic/Proper_classhttp://www.research.att.com/~njas/sequences/A002416http://www.enotes.com/topic/On-Line_Encyclopedia_of_Integer_Sequenceshttp://www.enotes.com/topic/Transitive_relationhttp://www.enotes.com/topic/Transitive_relationhttp://www.enotes.com/topic/Reflexive_relationhttp://www.enotes.com/topic/Reflexive_relationhttp://www.enotes.com/topic/Preorderhttp://www.enotes.com/topic/Preorderhttp://www.enotes.com/topic/Partially_ordered_sethttp://www.enotes.com/topic/Partially_ordered_sethttp://www.enotes.com/topic/Weak_orderhttp://www.enotes.com/topic/Weak_orderhttp://www.enotes.com/topic/Total_orderhttp://www.enotes.com/topic/Total_orderhttp://www.enotes.com/topic/Equivalence_relationhttp://www.enotes.com/topic/Equivalence_relationhttp://www.enotes.com/topic/OEIShttp://www.enotes.com/topic/OEIShttp://www.research.att.com/~njas/sequences/A002416http://www.research.att.com/~njas/sequences/A002416http://www.research.att.com/~njas/sequences/A006905http://www.research.att.com/~njas/sequences/A006905http://www.research.att.com/~njas/sequences/A053763http://www.research.att.com/~njas/sequences/A053763http://www.research.att.com/~njas/sequences/A000798http://www.research.att.com/~njas/sequences/A000798http://www.research.att.com/~njas/sequences/A001035http://www.research.att.com/~njas/sequences/A000670http://www.research.att.com/~njas/sequences/A000142http://www.research.att.com/~njas/sequences/A000110


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Notes:

The number of irreflexive relations is the same as that of reflexive relations. The number ofstrict partial orders (irreflexive transitive relations) is the same

as that of partial orders. The number of strict weak orders is the same as that of total preorders. The total orders are the partial orders that are also total preorders. The

number of preorders that are neither a partial order nor a total preorder is,therefore, the number of preorders, minus the number of partial orders,minus the number of total preorders, plus the number of total orders: 0, 0, 0,3, and 85, respectively.

the number of equivalence relations is the number ofpartitions, which is theBell number.

The binary relations can be grouped into pairs (relation,complement), except that forn = 0 therelation is its own complement. The non-symmetric ones can be grouped into quadruples

(relation, complement, inverse, inverse complement).

Binary relations by property

reflexi

ve

symmet

ric

transiti

vesymbol example

directed graph

undirected

graphNo Yes

tournament No Nopecking

order

dependency Yes Yes

weak order Yes

preorder Yes Yes preference

partial order Yes No Yes subset

partial

equivalenceYes Yes

equivalence Yes Yes Yes , , , equality
http://www.enotes.com/topic/Partially_ordered_set#Strict_and_non-strict_partial_ordershttp://www.enotes.com/topic/Partition_of_a_sethttp://www.enotes.com/topic/Bell_numberhttp://www.enotes.com/topic/Binary_relation#Complementhttp://www.enotes.com/topic/Binary_relation#Complementhttp://www.enotes.com/topic/Quadruplehttp://www.enotes.com/topic/Binary_relation#Operations_on_binary_relationshttp://www.enotes.com/topic/Reflexive_relationhttp://www.enotes.com/topic/Reflexive_relationhttp://www.enotes.com/topic/Symmetric_relationhttp://www.enotes.com/topic/Symmetric_relationhttp://www.enotes.com/topic/Transitive_relationhttp://www.enotes.com/topic/Transitive_relationhttp://www.enotes.com/topic/Directed_graphhttp://www.enotes.com/topic/Graph_(mathematics)http://www.enotes.com/topic/Graph_(mathematics)http://www.enotes.com/topic/Tournament_(graph_theory)http://www.enotes.com/topic/Pecking_orderhttp://www.enotes.com/topic/Pecking_orderhttp://www.enotes.com/topic/Dependency_relationhttp://www.enotes.com/topic/Weak_orderhttp://www.enotes.com/topic/Preorderhttp://www.enotes.com/topic/Preferencehttp://www.enotes.com/topic/Partial_orderhttp://www.enotes.com/topic/Subsethttp://www.enotes.com/topic/Partial_equivalence_relationhttp://www.enotes.com/topic/Partial_equivalence_relationhttp://www.enotes.com/topic/Equivalence_relationhttp://www.enotes.com/topic/Equality_(mathematics)http://www.enotes.com/topic/Partially_ordered_set#Strict_and_non-strict_partial_ordershttp://www.enotes.com/topic/Partition_of_a_sethttp://www.enotes.com/topic/Bell_numberhttp://www.enotes.com/topic/Binary_relation#Complementhttp://www.enotes.com/topic/Quadruplehttp://www.enotes.com/topic/Binary_relation#Operations_on_binary_relationshttp://www.enotes.com/topic/Reflexive_relationhttp://www.enotes.com/topic/Reflexive_relationhttp://www.enotes.com/topic/Symmetric_relationhttp://www.enotes.com/topic/Symmetric_relationhttp://www.enotes.com/topic/Transitive_relationhttp://www.enotes.com/topic/Transitive_relationhttp://www.enotes.com/topic/Directed_graphhttp://www.enotes.com/topic/Graph_(mathematics)http://www.enotes.com/topic/Graph_(mathematics)http://www.enotes.com/topic/Tournament_(graph_theory)http://www.enotes.com/topic/Pecking_orderhttp://www.enotes.com/topic/Pecking_orderhttp://www.enotes.com/topic/Dependency_relationhttp://www.enotes.com/topic/Weak_orderhttp://www.enotes.com/topic/Preorderhttp://www.enotes.com/topic/Preferencehttp://www.enotes.com/topic/Partial_orderhttp://www.enotes.com/topic/Subsethttp://www.enotes.com/topic/Partial_equivalence_relationhttp://www.enotes.com/topic/Partial_equivalence_relationhttp://www.enotes.com/topic/Equivalence_relationhttp://www.enotes.com/topic/Equality_(mathematics)


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relation

strict partial

orderNo No Yes

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