Introduction to Set theory and Logic (INTOSET) · Introduction to Set theory and Logic (INTOSET)...

Introduction to Set theory and Logic(INTOSET)

Relations,Partitions and Functions

Francis Joseph Campena,PhDMathematics and Statistics Department

De La Salle University-Manila

Relation

DefinitionLet A and B be sets. A binary relation from A to B is a subset ofA× B.

Most of the time when we talk about binary relation, we will justrefer to it as a relation.

This means that a binary relation from A to B is a set R ofordered pairs where the first element comes from A andthe second element comes from B.

Notation: The symbol aRb would denote that (a,b) ∈ Rand that the element a is ”related" the the element b.

EXAMPLEConsider the sets A = {0,1,2} and B = {a,b}. Then the set{(0,a), (0,b), (1,a), (2,b)} is a relation from A to B.

Relation

Relation on a Set

DefinitionA relation on a set A is a relation from A to A.

This means that a relation on a set A is a subset of A× A.

Example

Let A = {1,2,3,4}. If Ri is a relation on A, which of the orderedpairs belong to relation

1 R1 = {(a,b) : a divides b}.2 R2 = {(a,b) : a = b}.3 R3 = {(a,b) : a ≤ b}.4 R4 = {(a,b) : a > b}.

Relation on a Set

Example

Relation on a Set

Example

Relation on a Set

Example

Relation on a Set

Example

Examples

Consider the set A = {−2,−1,0,1,2} and the followingrelations

1 R1 = {(a,b) : a ≤ b}.2 R2 = {(a,b) : a > b}.3 R3 = {(a,b) : a = b or a = −b}.4 R4 = {(a,b) : a = b + 1}.5 R5 = {(a,b) : a + b ≤ 3}.

List all the elements of each relations given above.

Examples

Consider the set A = Z and the following relations1 R1 = {(a,b) : a ≤ b}.2 R2 = {(a,b) : a > b}.3 R3 = {(a,b) : a = b or a = −b}.4 R4 = {(a,b) : a = b + 1}.5 R5 = {(a,b) : a + b ≤ 3}.

Which of the relations above contain the pair1 (1,1)?2 (1,2)?3 (1,−1)?4 (−1,1)?5 (2,1)?

Examples

Properties

DefinitionConsider a set A and a relation R on A.

R is said to be reflexive if (a,a) ∈ R for every a ∈ A.R is said to be symmetric if (a,b) ∈ R then (b,a) ∈ R fora,b ∈ A.R is said to be antisymmetric if (a,b) ∈ R then (b,a) /∈ Rfor a,b ∈ A.R is said to be transitive if (a,b) ∈ R and (b, c) ∈ R then(a, c) ∈ R for a,b, c ∈ A.

Examples

Which of the following relations are1 reflexive?2 symmetric?3 antisymmetric?4 transitive?

EXERCISES

Determine whether the relation R on the set of all people isreflexive, symmetric, antisymmetric, and/or transitive, where(a,b) ∈ R if and only if

1 a is taller than b.2 a and b were born on the same day.3 a has the same first name as b.

REFLECT: Think of another relation on the set of all peoplewhich is reflexive, symmetric and transitive.

EXERCISES

Determine whether the relation R on the set of all people isreflexive, symmetric, antisymmetric, and/or transitive, where(a,b) ∈ R if and only if

1 a is taller than b.2 a and b were born on the same day.3 a has the same first name as b.

REFLECT: Think of another relation on the set of all peoplewhich is reflexive, symmetric and transitive.

Equivalence Relation

DefinitionA relation R on a set A is called an equivalence relation if it isreflexive, symmetric and transitive.

ExampleSuppose R is a relation on the set of strings of English letterssuch that aRb if and only if l(a) = l(b) where l(x) is the lengthof the string x . Is R an equivalence relation?

ExampleLet R be a relation on the set of real number such that aRb ifand only if a− b is an integer. Is R an equivalence relation?

Equivalence Relation

DefinitionA relation R on a set A is called an equivalence relation if it isreflexive, symmetric and transitive.

ExampleSuppose R is a relation on the set of strings of English letterssuch that aRb if and only if l(a) = l(b) where l(x) is the lengthof the string x . Is R an equivalence relation?

ExampleLet R be a relation on the set of real number such that aRb ifand only if a− b is an integer. Is R an equivalence relation?

Examples

DefinitionLet m be a positive integer greater than 1. If a and b areintegers, then a is congruent to b modulo m if m divides a− b.We use the notation a ≡ b mod m to indicate that a iscongruent to b modulo m.

Show that if A = Z and R = {(a,b) : a ≡ b mod m}, then R isan equivalence relation.

Examples

DefinitionLet m be a positive integer greater than 1. If a and b areintegers, then a is congruent to b modulo m if m divides a− b.We use the notation a ≡ b mod m to indicate that a iscongruent to b modulo m.

Show that if A = Z and R = {(a,b) : a ≡ b mod m}, then R isan equivalence relation.

Equivalence Class

DefinitionLet R be an equivalence relation of a set A. The set of allelements that are related to an element a ∈ A is called theequivalence class of a.

The equivalence class of a with respect the the relation R isdenoted by [a]R. When only one relation is being consideredand is clear from the discussion, we omit the subscript R andwrite [a] the equivalence class of a.If b ∈ [a]R, we call b a representative of this equivalence class.

Examples

1 Let A = Z. Let R be the relation such that aRb if and only ifa = b or a = −b.What is the equivalence class on aninteger for this equivalence relation?

2 What is the equivalence class of 0 for the congruencemodulo 4 on the set of all integers?

3 What are the equivalence classes of 0 and 3 for thecongruence modulo 5 on the set of all integers?

4 Consider the set A = {1,2,3,4,5,6,7,8,9,10}. Let R bethe relation aRb if and only if a ≡ b mod 5. What are thedistinct equivalence classes for this relation?

Examples

Parititions

DefinitionA paritition of a set S is a collection of disjoint non-emptysubsets of S whose union is the set S.

Example

Consider S = {1,2,3,4,5,6}. The collection of setsA1 = {1,3,5},A2 = {2,4},A3 = {6} forms a partition of S.

Can you give other partitions of S?

Paritions

TheoremLet R be an equivalence relation of a set S. Then theequivalence classes of R form a partition of S. Conversely,given a partition {Ai : i ∈ I} of the set S, there is an equivalencerelation R that has the sets Ai , i ∈ I as its equivalence classes.

ExampleWhat are the sets in the partition of the intgers arising from thecongruence modulo 4?

Functions

DefinitionA function f from a set X into a set Y is a rule that assigns toeach x in X a unique element f (x) in Y . The collection G ofpairs of the form (x , f (x)) in X × Y is called the graph of thefunction f .

A subset G of X × Y is the graph of a function on X if andonly if for each x ∈ X there is a unique pair in G whose firstelement is x .The word mapping is often used as a synonym forfunction.We express the fact that f is a function from X to Y bywriting f : X 7→ Y . The set X is the domain and Y tocodomain

Functions

The set Im(f ) = f [X ] = {y ∈ Y : y = f (x)} is called therange of f .If B is a subset of Y , we define the inverse image f−1[B] ofB to be the set of those x ∈ X such that f (x) ∈ B.

Let A ⊂ X . By the set f [A] = {y ∈ A : y = f (x)} we meanthe image of the set A under the function f .A function f : X 7→ Y is called one-to-one or injective iff (x1) = f (x2) implies that x1 = x2.

If a function f : X 7→ Y has a range equal to Y , we call thefunction an onto function or a surjective function.A function that is both one-to-one and onto is called a 1-1correspondence or a bijective function.

Examples

Identify whether the following function is 1− 1, onto or abijective function.

f (x) = x + 1f (x) = x2

f (x) =√

4− x2

f (x) = |x |

RECALL: An indexed subset of X or a collection of subsets ofX is a function on an index set S to X .⋃

As = {x ∈ X : ∃s(s ∈ S and x ∈ As)}⋂s∈S

As = {x ∈ X : ∀s(s ∈ S implies x ∈ As)}

If f : X 7→ Y and {Ak} is a collection of subsets of X , then

[⋃s∈S

⋃s∈S

f [As]

[⋂s∈S

⋂s∈S

f [As]

[⋃s∈S

⋃s∈S

f−1 [As]

[⋂s∈S

⋂s∈S

f−1 [As]

Introduction to Set theory and Logic (INTOSET) · Introduction to Set theory and Logic (INTOSET)...

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