GATE CSE 2000 | Question: 2.7

Question

Let $a, b, c, d$ be propositions. Assume that the equivalence $a ⇔ ( b \vee \neg b)$ and $b ⇔c$ hold. Then the truth-value of the formula $(a ∧ b) → (a ∧ c) ∨ d$ is always

• A. True
• B. False
• C. Same as the truth-value of $b$
• D. Same as the truth-value of $d$

Comments

$a \leftrightarrow (b \lor \neg b) = a \leftrightarrow T \Rightarrow a = T$

$\therefore (a \land b) \rightarrow (a \land c) \lor d = b \rightarrow c \lor d = \neg b \lor b \lor d = T$

Ans: (A.)

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Answer 1

Given  that $\ a\Leftrightarrow (b\vee \neg b)$ and $ b\Leftrightarrow c$
Now,
$(a\wedge b)\rightarrow (a \wedge c)\vee d$
$\equiv (a\wedge b)\rightarrow (a \wedge b)\vee d\;\;\;\; (\because b \Leftrightarrow c)$
$\equiv \neg (a\wedge b)\vee (a \wedge b)\vee d$
$\equiv T \vee d$
$\equiv T$
Hence, Option(A) True.

Comments

o yes thanks ..i missunderstood

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It is given that a⇔(b∨∼b) which implies a⇔T so there’s nothing wrong if we put a as T right?

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Yes, we can use a=T and b=c.

Answer 2

Given that, ab∨~b

It is equivalent to aTRUE

(a∧b)((a∧c)∨d)

wkt, 1∧x = x

(a∧b) = 1∧b = b

similarly, 1∧c = c

We now have, b (c∨d)

Which can be written as,

~b∨c∨d

We also know that bc

~b∨c = TRUE

TRUE∨d = TRUE

And hence answer is option a

Comments

Explanation is correct, just to highlight the following statement

We also know that bc  holds

 ~b∨c = TRUE

if you mean  bc   means   ~b∨c = TRUE   then its not true. Although you may not mean this, but while reading the answer it seems that.

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Nice Explanations

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great explanation

Answer 3

a ⇔ ( b V ~b) = and a ⇔  True means both a and True are equivalent
 b ⇔c means both b and c are equivalent

(a ∧ b) → (a ∧ c) ∨ d
=(True ∧ b) → (True ∧ c) ∨ d     (a ⇔  True)
= b → c ∨ d
= ~b ∨ c ∨ d
= ~b ∨ b ∨ d     (b ⇔c)
= True  ∨ d
=True

Hence ans is A

Answer 4

This will be helpful. see my solution: 

Comments

Nyc.....

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Nice hand writing ...