Unformatted text preview: the purported identity isn't always true.) 5.2.4 The following is a proof that for all sets A and B, if A ⊆ B, then A ∪ B ⊆ B. Proof: Suppose A and B are any sets and A ⊆ B. [ We must show that A ∪ B ⊆ B. ] ] Let x ∈ A ∪ B. [ We must show that x ∈ B. ] By the definition of the union A ∪ B, either: (i) x ∈ A, or ii) x ∈ B. ● In the first case, x ∈ A, so by the definition of A ⊆ B, x ∈ B. ● In the second case, x ∈ B. Thus, in both cases, x ∈ B. We have shown in general that if x ∈ A ∪ B, then x ∈ B. Therefore, by the definition of subset, A ∪ B ⊆ B....
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 Spring '08
 Ellis,W
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