Sets Day 1 Part II

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 = {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 √ A: Yes, A ⊆ B.

4. Q: Is A a subset of B? Q: Is B a subset of A? A = {1, 2, 3} B = {2} A: No. A⊈B 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 • True • True • False • True • True • False • True 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}

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 • False • True • True • False; (a is not a set.) • True • True • True • False • True • True • True • {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 • Examples done in class. • If n(A)=k, then the number of subsets of A is .