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