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Lecture 3 Handout

Lecture 3 Handout - Example Proof 2 Prove that for any two...

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Lecture 3 Handout Example Proof 1 Prove that the set of all powers of 2 (beginning with 2) is a subset of the set of all even numbers Given: Two sets A and B, } , 2 | { + = = Z j where a a A j } , 2 | { + = = Z k where k b b B Statement Justification Let x be an arbitrary element of A . We can write j x 2 = Given Rewrite as 1 2 2 - = j x Factoring Again rewrite as + = Z m where m x , 2 Substitution B x Definition of B Since x is an arbitrary element, all elements of A must be B Definition of inclusion B A Definition of subset
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Unformatted text preview: Example Proof 2 Prove that for any two sets A and B that B A B A ∪ ⊆ ∩ Statement Justification Let x be an arbitrary element of B A ∩ Given B x A x ∈ ∈ and Definition of intersection Since A x ∈ , B A x ∪ ∈ Definition of union Since x is an arbitrary element, all elements of B A ∩ must be B A ∪ ∈ Definition of inclusion B A B A ∪ ⊆ ∩ Definition of subset...
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