Problems I: Mathematical statements and proofs

**1.** By using truth tables prove that, for all statements *P* and *Q*, the statement *‘P* ⇒ *Q’* 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) ‘*P* ⇒ *Q*’ (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 ...

Start Free Trial

No credit card required