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Unit 1.2
SET RELATIONS
LEARNING OBJECTIVES
Upon completion you should be
able to determine set relations
such as
equality
subset and superset
equivalence
Two sets A and B are equal if
and only if they have the same
elements.
Equal Sets
In symbols, we write A=B.
Otherwise, we write A B.
Example 1.2.1
Let A = {1,2,3} and B = {x|x is a counting
number less than 4}.
Are all the elements in A also in B?
Are all the elements in B also in A?
YES! Therefore A and B have the same elements.
Sets A and B are _______.
Let C = {x|x is a letter in the word MATH}
and D = {A, T, M, H}.
Are all the elements in C also in D?
Are all the elements in D also in C?
YES! Therefore C = D.
Example 1.2.2
Remarks:
1. In a set, an element should not be
listed more than once.
2. In a set, the order of listing the
elements does not matter.
3. Two sets are equal if and only if
they have the same elements.
If all elements of set A are also
elements of set B, we say, A is a
subset of B or B is a superset of A.
Subset
In symbols, we write A B.
If A is not a subset of B,
we write A B.
Let S be the set of all students in this
room.
B be the set of all boys in this room.
G be the set of all girls in this room.
Is B a subset of S? Is G a subset of S?
Example 1.2.3
B S G S
Let N= {1,2,3,4,5,6,}
W={0,1,2,3,4,5,6,}
Example 1.2.4
Yes
No
Is N W?
Is W N?
Let A be a non-empty set.
If A B and A B, then we call A
a proper subset of B.
Proper Subset
In symbols, we write A B.
Given A ={, } and B = {,,,},
determine whether A is a proper
subset of B.
Since A is non-empty, A B and
A B then, A is a proper subset of
B and we write A B.
Example 1.2.5
Let N= {1,2,3,4,5,6,}
W={0,1,2,3,4,5,6,}
Example 1.2.6
Yes
No
Is N W?
Is W N?
TIME TO THINK!
True or False: Let A, B, and C be
sets. 1. A A.
2. A B then B A.
3. If A B and B C, then A C.
4. A
5.
6. A U
Finite and Infinite Sets
A set is finite if its elements
can be counted (and the
counting process is terminal).
Otherwise, the set is infinite.
N= {1,2,3,4} is finite.
Example 1.2.7
W={0,1,2,3,4,5,6,} is infinite.
Two sets A and B are in one-to-one
correspondence if it is possible to pair each
element of A with exactly one element of B,
and each element of B with exactly one
element of A.
When two sets A and B are in 1-1
correspondence, we say they are equivalent
and we write A B.
Equivalent Sets
A = {11,12, 13, 14, 15}
B = {ate, egg, irk, off, urn}
C = {strawberry, peach, apple}
D = {do, re, mi}
A B and C D. Is A C? Why?
Example 1.2.8
O = {,-3,-1,0,1,3,}
E = {,-4,-2,0,2,4}
Is O E? Why?
Example 1.2.9
TIME TO THINK!
1.Are all equal sets equivalent?
2.Are equivalent sets equal?
3.Can a set be equivalent to
any of its subsets?
4.Can a set be equal to any of
its subsets?
In this section, we learned
When two sets are equal;
When a set is a subset or superset
of another;
When two sets are equivalent sets.
Summary