Mathematical Reasoning Practice Questions - DPP for JEE

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6. (c) S (p, q, r) = ~ p ? [~ (q ? r)]
So,  S (~p, ~q, ~r) = ~ (~p) ? [~ (~q v ~r)] = p ? (q ? r)
S*(p, q, r) = ~ p ? [~ (q ? r)]
S* (~p, ~q, ~r) = p ? (q ? r)
Clearly, S* (~p, ~q, ~r) = ~ S (p, q, r)
7. (b)
8. (b)
9. (b)
10. (d) Since  (By De-Morgans’ law)
? 
? (d) is the false statement
11. (a) We know that the contropositive of p ? q  is
~ q ? ~ p. So contrapositive of p ? (~q ? ~r) is
~ (~q ? ~r) ? ~p
= ~ q ? [~ (~r)] ? ~p
[ Q ~ (p ? q) = p ? ~q]
= ~ q ? r ? ~p
12.  (d) The truth tables of p ? q and ~ p ? q are given below:
Clearly, truth tables of  p ? q and ~ p ? q are same.
So, p ? q is logically equivalent to ~ p ? q.
Hence, option (a) is correct.
If the truth value of p, q, r are T, F, T respectively, then the truth values
of p ? q and q ? r are each equal to T.
Therefore, the truth value of (p ? q) ? (q ? r) is T.
Hence, option (b) is correct.
We have, ~ (p ? q ? r)  ( ~ p ? ~ q ? ~ r)
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6. (c) S (p, q, r) = ~ p ? [~ (q ? r)]
So,  S (~p, ~q, ~r) = ~ (~p) ? [~ (~q v ~r)] = p ? (q ? r)
S*(p, q, r) = ~ p ? [~ (q ? r)]
S* (~p, ~q, ~r) = p ? (q ? r)
Clearly, S* (~p, ~q, ~r) = ~ S (p, q, r)
7. (b)
8. (b)
9. (b)
10. (d) Since  (By De-Morgans’ law)
? 
? (d) is the false statement
11. (a) We know that the contropositive of p ? q  is
~ q ? ~ p. So contrapositive of p ? (~q ? ~r) is
~ (~q ? ~r) ? ~p
= ~ q ? [~ (~r)] ? ~p
[ Q ~ (p ? q) = p ? ~q]
= ~ q ? r ? ~p
12.  (d) The truth tables of p ? q and ~ p ? q are given below:
Clearly, truth tables of  p ? q and ~ p ? q are same.
So, p ? q is logically equivalent to ~ p ? q.
Hence, option (a) is correct.
If the truth value of p, q, r are T, F, T respectively, then the truth values
of p ? q and q ? r are each equal to T.
Therefore, the truth value of (p ? q) ? (q ? r) is T.
Hence, option (b) is correct.
We have, ~ (p ? q ? r)  ( ~ p ? ~ q ? ~ r)
So, option, (c) is correct.
If p is true and q is false, then p ? q is true. Consequently,
~ (p ? q) is false and hence p ? ~ (p ? q) is false.
Hence, option (d) is wrong.
13. (b) is logically equivalent to 
 is a tautology but not a contradiction.
14. (c)
15. (a)
16. (b) means .
17. (c) The inverse of the proposition  (p ? ~ q) ? r is
~  (p ? ~ q) ? ~ r
= ~ p ? ~ (~q) ? ~ r
= ~ p ? q ? ~ r
18. (a) Let p, q and r be three propositions given by
p : x = 5, q : y = –2 and r : x – 2y = 9
Then, the given statement is (p ? q) ? r
Its contrapositive is
~ r ? ~ (p ? q)
i.e., ~ r ? ~ p ? ~ q
i.e., If x – 2y ? 9, then x ? 5 or y ? –2
19. (b) Let us make the truth table for the given statements, as follows :
From table we observe
p ? (q?p) is equivalent to p?(p?q)
20. (c) ~ [ ( p ? q) ? (q ? ~ r)]  ~ ( p ? q) ? ~ (q ? ~ r)
 (~ p ? ~ q) ? (~ q ? r)
21. (b)
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6. (c) S (p, q, r) = ~ p ? [~ (q ? r)]
So,  S (~p, ~q, ~r) = ~ (~p) ? [~ (~q v ~r)] = p ? (q ? r)
S*(p, q, r) = ~ p ? [~ (q ? r)]
S* (~p, ~q, ~r) = p ? (q ? r)
Clearly, S* (~p, ~q, ~r) = ~ S (p, q, r)
7. (b)
8. (b)
9. (b)
10. (d) Since  (By De-Morgans’ law)
? 
? (d) is the false statement
11. (a) We know that the contropositive of p ? q  is
~ q ? ~ p. So contrapositive of p ? (~q ? ~r) is
~ (~q ? ~r) ? ~p
= ~ q ? [~ (~r)] ? ~p
[ Q ~ (p ? q) = p ? ~q]
= ~ q ? r ? ~p
12.  (d) The truth tables of p ? q and ~ p ? q are given below:
Clearly, truth tables of  p ? q and ~ p ? q are same.
So, p ? q is logically equivalent to ~ p ? q.
Hence, option (a) is correct.
If the truth value of p, q, r are T, F, T respectively, then the truth values
of p ? q and q ? r are each equal to T.
Therefore, the truth value of (p ? q) ? (q ? r) is T.
Hence, option (b) is correct.
We have, ~ (p ? q ? r)  ( ~ p ? ~ q ? ~ r)
So, option, (c) is correct.
If p is true and q is false, then p ? q is true. Consequently,
~ (p ? q) is false and hence p ? ~ (p ? q) is false.
Hence, option (d) is wrong.
13. (b) is logically equivalent to 
 is a tautology but not a contradiction.
14. (c)
15. (a)
16. (b) means .
17. (c) The inverse of the proposition  (p ? ~ q) ? r is
~  (p ? ~ q) ? ~ r
= ~ p ? ~ (~q) ? ~ r
= ~ p ? q ? ~ r
18. (a) Let p, q and r be three propositions given by
p : x = 5, q : y = –2 and r : x – 2y = 9
Then, the given statement is (p ? q) ? r
Its contrapositive is
~ r ? ~ (p ? q)
i.e., ~ r ? ~ p ? ~ q
i.e., If x – 2y ? 9, then x ? 5 or y ? –2
19. (b) Let us make the truth table for the given statements, as follows :
From table we observe
p ? (q?p) is equivalent to p?(p?q)
20. (c) ~ [ ( p ? q) ? (q ? ~ r)]  ~ ( p ? q) ? ~ (q ? ~ r)
 (~ p ? ~ q) ? (~ q ? r)
21. (b)
22. (a)
 is a tautology.
23. (c) (p  q)  (p  r)
 (T  T)  (T  F)
 T  T
 T
24. (c) Consider ~ [p ? (q ? r)] = ~p ? ~ (q ? r)
= ~p ? (~q ? ~ r)
= (~p ? ~q) ? (~p ? ~ r)
25. (a) Suman is brilliant and dishonest if and only if Suman is rich is
expressed as
Negation of it will be