Sets Day 1 Part II

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.

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.

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

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.

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

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.

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.)

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