Logic (Mathematical Reasoning) (Solved MCQs and Notes)

Notes 4 Pages

Joint Entrance Examination (Main)

Mathematics

Contributed by

Nikita Narasimhan

305
Unit - 16
Mathematical Reasoning
Summary
1. ~


p p

2. ~
     
p q p V q
   
~
     
pV q p q
  
3.


p q p V q
  

q p  
4.
p q q p
  

   
p q q p
   





p V q q V p
  
5. ~
     
p q p q V q p
    
p q
  
p q
 
6.
pVq q V p
p q q p



  





   
p Vq Vr p V q Vr
p q r p q r



    


7. Tautology : The statement which is always true is called tautology is denoted by t.
     
p q V r p q V p r
   
     
p V q r p Vq p Vr
  
8. Cantradiction or fallacy
The statement which is always false is called contradiction.
is denoted by 'c' or 'f '.
(i)
p V t t

(ii)
p t p
 
(iii)


p V p t

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9. Contrapositive of
p q is q p
 
10.
q
p

is false only when p is true and q is false
QUESTION BANK
1.
   
p q p q
   
is
(a) a contradiction (b) a tautology
(c) neither a tautology nor a contradiction (d) both tautology and contradiction
2. Which of the following a tautology ?
(a)


p p
 
(b)
p c

(c)
p V t
(d)
p p

3. Which of the following is true ?
(a)


p p t
 
(b)
 
p V p f

(c)
p q q p
  
(d)
   
P q q p   
4. If both p and q are false then
(a)
p q

is true (b)
p V q
is false
(c)
q
p

is false (d)
 
p V q

is false
5. If both p and q are true
(a)
p q

is true (b)
p V q
is false
(c)
q
p

is false (d) None of them
6.
p q V r

is false then the true values of p, q and r are respectively..
(a) F, T, T (b) T, T, F (c) T, F, F (d) F, F, F
7. The logically equivalent proposition of
p q
 
is
(a)
q p
(b)
p q

(c)
q p 
(d)
p q
 
8. The contrapositive of the converse of

q
p

is
(a)


p q V p

(b)
   
p q q p
  
(c)
   
p q q p
  
(d)
   
p q p Vq
 
9.




p V q V p q
 
= . . . . . . . . . . . . . . . .
(a) q (b) p (c) ~ p (d) ~ q
10. The proposition of
   
p p p p
   
is
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(a) neither tautology nor contradiction (b) contradiction
(c) Tautology and contradiction (d) tautology
11. Each of the following statement is true
p q

p r
 
~ r Then
(a) p is false (b) p is true
(c) q is true (d) None of these
12. If each of the statement
p q
 
and
r q
, p is true than
(a) r is false (b) r is true
(c) q is true (d) None of these
13. The negation of compound proposition
 
p V p V q

is ........
(a)


p q p
  
(b)
 
p p V q
  
(c)
   
p q V p
  
(d)


p q V p
  
14.


p q p r  
is
(a) Contradiction (b) tautology
(c) Neither contradiction Nor tautology (d) Both contradiction & tautology
15. Negation of 'for all x , p’ is ........
(a) there exists x ,

p (b) for all x;

p
(c) ~P (d) P
16.
   
p q q p
   
is a
(a) contradiction (b) tautology
(c) both tautology & contradiction (d) None of above
17.
 
p q p
 
is equivalent to
(a)
 
qpp 
(b)
 
qpp 
(c)


p p V q

(d)


p p q
 
18. If p = He is intelligent
q = He is strong
Then symolic form of the statemant
" It is wrong that he is intelligent or strong " is
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308
(a)
p V q
 
(b)
 
p q

(c)
 
p V q

(d)
p V q

19. If statement p and r are false and q is true then truth value of
 
p q r V r
 
is .........
(a) T (b) F
(c) T or F (d) Can not say
20. Which one of the following is false
(a)
 
p p
 
is a contradiction (b)
   
p q q p
   
is a contradiction
(c)
 
p p
 
is a tautology (d)
 
p V p

is a tautology
21.
 
p p V q
 
is false then value of p and q are respectively
(a) F, F (b) F,T (c) T, T (d) T, F
ANSWERS
1 a 11 a 21 d
2 c 12 b
3 d 13 a
4 b 14 a
5 a 15 a
6 c 16 d
7 b 17 c
8 b 18 c
9 c 19 b
10 b 20 b
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