Introductory Chapter : Mathematical Logic, Proof and Sets
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Exercise I(e)
1. Construct a truth table for  P   Q .
2. Show that
(a)  P   P is a tautology.
(b)  P   P is a contradiction.
3. Construct a truth table for
 P   Q     R   R      P  Q 
What do you notice about your results?
4. Show that the following are tautologies:
(a)  P  Q    P  R     P   Q  R  
(b)  P  Q    R  Q     P  R   Q 
5. State the contrapositive and converse of
(a) If x is even then x 2 is even
(b) If x 2 is odd then x is odd
6. Let x and y be real numbers. Let P be ‘ x  y ’ and Q be ‘ x 2  y 2 ’. Write the
following propositions in words and state which ones are true.
(i) P  Q
(ii) The converse of P  Q
(iii) The contrapositive of P  Q
7. Show that
 P  Q    Q     P 
is a tautology.
8. Determine
(i) The converse of P   Q 
(ii) The contrapositive of P   Q 
(iii) Converse of the contrapositive of  P    Q 
9. Give an example of where P  Q is true but the converse Q  P is false.
10. State the contrapositive and converse of:
If b 2  4ac  0 then the equation ax 2  bx  c  0 has real roots.
Also state whether the converse is true or false.
Introductory Chapter : Mathematical Logic, Proof and Sets
11. State the contrapositive and converse of:
n
If 1  a  0 then 1  a   1  na
[This is Bernoulli’s Inequality]
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