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An Introduction to Mathematical Reasoning by Peter J. Eccles

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Problems I: Mathematical statements and proofs

1. By using truth tables prove that, for all statements P and Q, the statement ‘PQ’ and its contrapositive ‘(not Q) ⇒ (not P)’ are equivalent. In Example 1.2.3 identify which statement is the contrapositive of statement (i) (f(a) = 0 ⇒ a > 0). Find another pair of statements in that list which are the contrapositives of each other.

2. By using truth tables prove that, for all statements P and Q, the three statements (i) ‘PQ’ (ii) ‘(P or Q) ⇔ Q’ and (iii) ‘(P and Q) ⇔ P’ are equivalent.

3. Prove that the three basic connectives ‘or’, ‘and’ and ‘not’ can all be written in terms of the single connective ‘notand’ where ‘P notand Q’ is interpreted as ‘not (P and Q)’.

4. Prove the following statements ...

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