1

### JEE Main 2016 (Online) 9th April Morning Slot

Consider the following two statements :

P :     If 7 is an odd number, then 7 is divisible by 2.
Q :    If 7 is a prime number, then 7 is an odd number

If  V1 is the truth value of the contrapositive of P and V2 is the truth value of contrapositive of Q, then the ordered pair (V1 , V2) equals :
A
(T, T)
B
(T, F)
C
(F, T)
D
(F, F)

## Explanation

Contrapositive of P : If 7 is not divisible by 2, then 7 is not an odd number.

This statement is false.

$\therefore$    V1 = False (F)

Contrapositive of Q : If 7 is not an odd number ,then 7 is not a prime number.

This statement is true.

$\therefore$   V2 = True (T)

$\therefore$   (V1, V2) = (F, T).
2

### JEE Main 2016 (Online) 10th April Morning Slot

The contrapositive of the following statement,

“If the side of a square doubles, then its area increases four times”, is :
A
If the side of a square is not doubled, then its area does not increase four times.
B
If the area of a square increases four times, then its side is doubled.
C
If the area of a square increases four times, then its side is not doubled.
D
If the area of a square does not increase four times, then its side is not doubled.

## Explanation

Contrapositive of p $\to$ q is $\sim$q $\to$ $\sim$p.

Here,

Let

p = Side of a square is doubles.

q = Area of square increases four times.

$\therefore$   $\sim$q $\to$ $\sim$p = If the area of a square does not increase four times, then its side is not doubled.
3

### JEE Main 2017 (Offline)

The following statement

$\left( {p \to q} \right) \to \left[ {\left( { \sim p \to q} \right) \to q} \right]$ is
A
equivalent to ${ \sim p \to q}$
B
equivalent to ${p \to \sim q}$
C
a fallacy
D
a tautology

## Explanation

We have

p q ~p p$\to$q ~p$\to$q (~p$\to$q)$\to$q (p$\to$q)$\to$((~p$\to$q)$\to$q)
T F F F T F T
T T F T T T T
F F T T F T T
F T T T T T T

$\therefore$ It is tautology.
4

### JEE Main 2017 (Online) 8th April Morning Slot

The proposition $\left( { \sim p} \right) \vee \left( {p \wedge \sim q} \right)$ is equivalent to :
A
p $\vee$ ~ q
B
p $\to$ ~ q
C
p $\wedge$ ~ q
D
q $\to$ p