1. Sets Day 1 Part II

2. Definition of Subset
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:
A is first and foremost a SET. (A can’t be a subset if it isn’t a set.)
If x ∈ A, then x ∈ B.

3. Q: Is A a subset of B? A: Yes, A ⊆ B.
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?
A: Yes, A ⊆ B.

4. Q: Is A a subset of B? A: No. A⊈B Q: Is B a subset of A? A: Yes. B⊆A

5. Fact about the empty set.
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.

6. True or False {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
False

7. Definition of Proper Subset
Set A is a proper subset of set B, symbolized by A ⊂ B, if and only if the following three conditions hold:
A is a set.
Every element of A is also an element of B.
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.

8. True or False {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
False
True
False; (a is not a set.)

9. Number of Subsets of a Set
Examples done in class.
If n(A)=k, then the number of subsets of A is .