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+14 votes

Choose the correct alternatives (more than one may be correct) and write the corresponding letters only:

Which of the following is/are a tautology?

  1. $a \vee b \to b \wedge c$
  2. $a \wedge b \to b \vee c$
  3. $a \vee b \to \left(b \to c \right)$
  4. $a \to b \to \left(b \to c \right)$
asked in Mathematical Logic by Veteran (52k points)
edited by | 1k views

2 Answers

+15 votes
Best answer

Answer: (B)

$\left(a \wedge b \right) \to b \vee c$
$\implies \neg \left(a \wedge b \right) \vee b \vee c$
$\implies \neg a \vee \neg b \vee b \vee c$
$\implies T$

Option (A) is not TRUE when C is FALSE
Option (C) is not TRUE when b is TRUE and C is FALSE
Option (D) is not TRUE when a and b are TRUE and C is FALSE.

answered by Boss (33.8k points)
edited by
When I am solving option (b) , after simplification , I get  (~a +1+ c ) . So , for a=1 , b=0 , c=0 , output is 1.

and when I solve option (d) , after simplification I get (~a+~b+c) .

So , option (b) is tautology
Tautology means after simplification you should get TRUE.
ohkk , my mistake.. sorry .. Thanks a lot.

How to treat option D? (the associtivity)  a->(b->(b->c)) or (a->b)->(b->c)

Is not option (d) ambiguous?
Take it either way - both are not tautology for option D.
@Arjun any of the associativity work for this problem. But in general what is the associativity?

Associativity like


  • AND,OR is left associative .
  • Implecation is right associative
+5 votes

Option b is tautology.

answered by Boss (33.9k points)


Nice approach

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