PGEE 2018

Question

p: n is a prime number

q: n mod 30 is prime number

1) p implies q

2) q implies p

3) p implies q or q implies p

4) If p implies q then q implies p

Answer 1

Option C is correct. 

p: n is a prime number
q: n mod 30 is a prime number

Option A - 
$p\rightarrow q$
Statement - If n is a prime number then n mod 30 is a prime number
Counter Example - let n = 31
If 31 is a prime number then 31 mod 30 is a prime number
31 mod 30 = 1
1 is not a prime number.
$p\rightarrow q$ = $T\rightarrow F $  = F

Options B and D are same.
B is $ q\rightarrow p$
Option D-
($p\rightarrow q$)  $\rightarrow$ ($ q\rightarrow p$)
= $\neg$($\neg p$ $\vee q$) $\vee$ ($\neg q$ $\vee p$)
=($p \wedge \neg q $) $\vee$ ($\neg q$ $\vee p$)
=(($p \wedge \neg q $) $\vee$ $\neg q$)  $\vee$ (($p \wedge \neg q $)  $\vee p$)
=$\neg q$ $\vee p$ 
= $ q\rightarrow p$
Statement - If n mod 30 is a prime number then is a prime number
Counter-example for options B and D - 
Take n = 32
n mod 30 = 2 which is a prime number but n is not a prime number.
$q\rightarrow p$ = $T\rightarrow F $ = F 

Option C -
($p\rightarrow q$)  $\vee$ ($ q\rightarrow p$)
Now see.
If $p\rightarrow q$ is false then $q\rightarrow p$ must be true and vice versa. 
($p\rightarrow q$)  $\vee$ ($ q\rightarrow p$)
($\neg p$ $\vee q$) $\vee$ ($\neg q$ $\vee p$)
T
It's a tautology so it will always be true irrespective the value of n. 

Answer 2

Option 1 is wrong

because 31 = 1 mod 30 and 1 is not a prime number.

Option 2 is wrong

because 49 = 19 mod 30 here 19 is a prime number but 49 is not.

Lets see about option 3

The statement of option 3 is

P implies Q or Q implies P

to make this statement false we have to make both sides of OR false.

So,lets try to make P implies Q false.

To do that we have to make P true and Q false. If we do that the other side of or automatically gets true. So,option 3 is always true.

Now lets see about option 4 the statement of option 4 is

IfP implies Q then Q implies P

Note that this statement is wrong when P implies Q is true and Q implies P is false

We can make P implies Q true by making P false and Q true

Take n=49 then P is false but Q is true thus P implies Q is true but Q implies P becomes false.

So,option 4 is not correct.

Thus option 3 is the solution.

Comments

We already know 1 is not prime number then what is use of doing 1 mod 30? If "n" is prime then only you should perform "n mod 30" so  "1 mod 30" will give 1 as answer which is not prime but the condition is if "n" is a prime number then only "n mod 30" will give a prime number. so p implies q seems correct here.