Page 31 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

The same idea is useful for sets: if we have two sets A and B , and we want to show they are both the

same set, we can show

A ⊆ B,

and then show

B ⊆ A.

Proposition 3.1.3 (Subset Proposition). Let A and B be sets. If A ⊆ B and B ⊆ A, then A = B .

Proof: We will show that every element of A is also an element of B , and every element of B is also
an element of A. Then, A and B will necessarily be the same set, because they contain exactly the
same elements. Let a ∈ A . By the definition of A ⊆ B , this means a ∈ B . Now, let b ∈ B. By the
definition of B ⊆ A, this means b ∈ A. So every element is of A is also an element of B, and every
element of B is also an element of A. Therefore, A and B contain exactly the same elements, so they
are the same set.

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