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1

### JEE Main 2021 (Online) 27th August Evening Shift

The Boolean expression (p $$\wedge$$ q) $$\Rightarrow$$ ((r $$\wedge$$ q) $$\wedge$$ p) is equivalent to :
A
(p $$\wedge$$ q) $$\Rightarrow$$ (r $$\wedge$$ q)
B
(q $$\wedge$$ r) $$\Rightarrow$$ (p $$\wedge$$ q)
C
(p $$\wedge$$ q) $$\Rightarrow$$ (r $$\vee$$ q)
D
(p $$\wedge$$ r) $$\Rightarrow$$ (p $$\wedge$$ q)

## Explanation

given statement says

"if p and q both happen then p and q and r will happen"

it simply implies "If p and q both happen then 'r' too will happen"

i.e.

"if p and q both happen then r and p too will happen

i.e.

(p $$\wedge$$ q) $$\Rightarrow$$ (r $$\wedge$$ p)
2

### JEE Main 2021 (Online) 27th August Morning Shift

The statement (p $$\wedge$$ (p $$\to$$ q) $$\wedge$$ (q $$\to$$ r)) $$\to$$ r is :
A
a tautology
B
equivalent to p $$\to$$ $$\sim$$ r
C
a fallacy
D
equivalent to q $$\to$$ $$\sim$$ r

## Explanation

(p $$\wedge$$ (p $$\to$$ q) $$\wedge$$ (q $$\to$$ r)) $$\to$$ r

$$\equiv$$ (p $$\wedge$$ ($$\sim$$ p $$\vee$$ q) $$\vee$$ ($$\sim$$ q $$\vee$$ r)) $$\to$$ r

$$\equiv$$ ((p $$\wedge$$ q) $$\wedge$$ ($$\sim$$ p $$\vee$$ r)) $$\to$$ r

$$\equiv$$ (p $$\wedge$$ q $$\wedge$$ r) $$\to$$ r

$$\equiv$$ $$\sim$$ (p $$\wedge$$ q $$\wedge$$ r) $$\vee$$ r

$$\equiv$$ ($$\sim$$ p) $$\vee$$ ($$\sim$$ q) $$\vee$$ ($$\sim$$ r) $$\vee$$ r

$$\Rightarrow$$ tautology
3

### JEE Main 2021 (Online) 26th August Evening Shift

Consider the two statements :

(S1) : (p $$\to$$ q) $$\vee$$ ($$\sim$$ q $$\to$$ p) is a tautology .

(S2) : (p $$\wedge$$ $$\sim$$ q) $$\wedge$$ ($$\sim$$ p $$\wedge$$ q) is a fallacy.

Then :
A
only (S1) is true.
B
both (S1) and (S2) are false.
C
both (S1) and (S2) are true.
D
only (S2) is true.

## Explanation

S1 : ($$\sim$$ p $$\vee$$ q) $$\vee$$ (q $$\vee$$ p) = (q $$\vee$$ $$\sim$$ p) $$\vee$$ (q $$\vee$$ p)

S1 = q $$\vee$$ ($$\sim$$ p $$\vee$$ p) = qvt = t = tautology

S2 : (p $$\wedge$$ $$\sim$$ q) $$\wedge$$ ($$\sim$$ p $$\vee$$ q) = (p $$\wedge$$ $$\sim$$ q) $$\wedge$$ $$\sim$$ (p $$\wedge$$ $$\sim$$ q) = C = fallacy
4

### JEE Main 2021 (Online) 26th August Morning Shift

If the truth value of the Boolean expression $$\left( {\left( {p \vee q} \right) \wedge \left( {q \to r} \right) \wedge \left( { \sim r} \right)} \right) \to \left( {p \wedge q} \right)$$ is false, then the truth values of the statements p, q, r respectively can be :
A
T F T
B
F F T
C
T F F
D
F T F

## Explanation

p q r $$\underbrace {p \vee q}_a$$ $$\underbrace {q \to r}_b$$ $${a \wedge b}$$ $${ \sim r}$$ $$\underbrace {a \wedge b \wedge ( \sim r)}_c$$ $$\underbrace {p \wedge q}_d$$ $$c \to d$$
T F T T T T F F F T
F F T F T F F F F T
T F F T T T T T F F
F T F T F F T F F T

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