Sets Day 1 Part II

Set A is a subset of set B, symbolized A ⊆ B, if and only if all the elements of set A are also elements of set B.

Note that A is a subset of set B if the following two conditions hold:

1. A is first and foremost a SET. (A can’t be a subset if it isn’t a set.)

2. If x ∈ A, then x ∈ B.

Definition of Subset

A = {a, e, i, o, u}B = {letters in the English alphabet}

Check the conditions:1 – Is A a set? Yes √2 – Are the letters a, e, i, o, and u contained in set B? Yes √

Q: Is A a subset of B?

A: Yes, A ⊆ B.

A = {1, 2, 3} B = {2}

Q: Is B a subset of A?

A: No. A⊈B

Q: Is A a subset of B?

A: Yes. B⊆A

Fact: The empty set is a subset of every set.

Why? The reasoning is kind of hard to follow because you have to look at why it is that ɸ cannot not be a subset of every set.

Suppose that there is some set A of which ɸ is not a subset. Then that means that there is something in ɸ which is not in A. Since this can’t happen no such set A exists.

Fact about the empty set.

{1,2,3} = {3,2,1}

{1,2,3} ⊆ {3,2,1}

1∈ {1,2,3}

1⊆ {1,2,3}

{1} ⊆ {1,2,3}

ɸ⊆ {1,2,3}

ɸ∈ {1,2,3}

ɸ∈ {ɸ,{1,2,3},Fred}

True or False

True

True

True

False

True

True

False

True

Set A is a proper subset of set B, symbolized by A ⊂ B, if and only if the following three conditions hold:

1. A is a set.2. Every element of A is also an element of B.3. A ≠ B.

Note: The first two conditions imply that A must be a subset of B. Therefore A is a proper subset of B if A is a subset of B and A is not equal to B.

Definition of Proper Subset

{1,2,3} ⊂ {1,2,3}

{1,2} ⊂ {1,2,3} φ ⊂ {1,2,3} a ⊂ {a,b,c} a ∈ {a,b,c} {a} ⊂ {a,b,c} {1} ⊄ {1} φ ⊂ φ φ ⊆ φ φ = φ {0} ⊄ φ

True or False

False True True False; (a is not a set.) True True True False True True True

Examples done in class.

If n(A)=k, then the number of subsets of A is

Number of Subsets of a Set

k2 .