1. Sets 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 = {a, e, i, o, u} B = {letters in the English alphabet} Check the conditions: 1 – Is A a set? 2 – Are the letters a, e, i, o, and u contained in set B? A:

4. Q: Is A a subset of B? A: Q: Is B a subset of A? A:
A = {1, 2, 3} B = {2} A:
Q: Is B a subset of A? 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

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

9. Number of Subsets of a Set
List all the subsets of the set A = {1,2}
List all the subsets of the set B = {1,2,3}
List all the subsets of the set C = {a,b,c}
How many subsets will the set D have if D = {x,y,z}
How many subsets will the set E have if E = {1,2,3,4}
If n(A)=k, then the number of subsets of A is .