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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. $a \to b \to \left(b \to c \right)$

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

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.

edited
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

Not>AND>OR>IMPLICATION

• AND,OR is left associative .
• Implecation is right associative

Associativity like

Not>AND>OR>IMPLICATION

You mean Precedence of the Operator rt?

@arjun sir can you please telll me precedence of these operations

Option b is tautology .

@abhishekmehta4u

Nice approach

@abhishekmehta4u your approcah is good and all, but a minor mistake in the (A) part :

You accidentally wrote the R.H.S  of (A) as ($b \vee c$ )instead of ($b \wedge c$)