1. (a) Each of the following true statements contains an implication. (i) For all a, b E Z, if a – b is even then a + b is even. (ii) For all A C N, if A is finite then Ac is infinite. Write down the converse of each statement and decide (with justifica- tion) whether it is true or false. (b) Consider the statement: For all r, y, z E Z. At least one of x - y, x- z and y -z is even. (i) Write down a roadmap for a proof of this statement by contradic- tion. (ii) Fill in the details of your roadmap to prove the statement.

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ISBN:9780470458365
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1.
(a) Each of the following true statements contains an implication.
(i) For all a, b E Z, if a – b is even then a + b is even.
(ii) For all A C N, if A is finite then Aº is infinite.
Write down the converse of each statement and decide (with justifica-
tion) whether it is true or false.
(b) Consider the statement: For all x, y, z Z. At least one of x – y, x – %
and y – z is even.
(i) Write down a roadmap for a proof of this statement by contradic-
tion.
(ii) Fill in the details of your roadmap to prove the statement.
Transcribed Image Text:1. (a) Each of the following true statements contains an implication. (i) For all a, b E Z, if a – b is even then a + b is even. (ii) For all A C N, if A is finite then Aº is infinite. Write down the converse of each statement and decide (with justifica- tion) whether it is true or false. (b) Consider the statement: For all x, y, z Z. At least one of x – y, x – % and y – z is even. (i) Write down a roadmap for a proof of this statement by contradic- tion. (ii) Fill in the details of your roadmap to prove the statement.
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