GATE CSE 1992 | Question: 02,xvi

Question

Which of the following is/are a tautology?

• A. $a \vee b \to b \wedge c$
• B. $a \wedge b \to b \vee c$
• C. $a \vee b \to \left(b \to c \right)$
• D. $a \to b \to \left(b \to c \right)$

Comments

@Psy Duck don’t you think...proper parenthesis should have been given for option 4 ?

 

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Yes it should have but by default we take implication as right associative

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quite easy!!

Answer 1

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.

Comments

Associativity like

Not>AND>OR>IMPLICATION 

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

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Associativity like

Not>AND>OR>IMPLICATION 

You mean Precedence of the Operator rt? 

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

Answer 2

Option b is tautology.

Comments

@abhishekmehta4u

Nice approach

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@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 ($b \wedge c$)

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 abhishekmehta4u Option A is $a \lor b\rightarrow b\land c$,

in option C) $\bar a \bar b+\bar b+c$ should be $\bar b+c$

Answer 3

Answer: (B).

Lets go over each of the options with case method where $b = True$.

A)     $a \vee b → b \wedge  c$

$=> (a \vee True) → (True \wedge  c)$

$=> (True) → (c)$

$=> c$

Option A will not be Tautology if $c=False$.

 

B)     $a \wedge  b → b \vee c$

$=> (a \wedge True) → (True \vee c)$

$=> (a) → True$

$=> True$ (Anything implies True is always True)

Option B is Tautology.

 

C)     $a \vee b → (b → c)$

$=> a \vee True → (True → c)$

$=> (True) → (c) $

$=> c $

Option C will not be Tautology if $c=False$.

 

C)     $a →  b → (b → c)$ 
$=> a →  (True → (True → c))$

$=> a →  (True → (c))$

$=> a →  c$

Option D will not be Tautology if $a=True$ and $c=False$

Answer 4

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