GATE CSE 2016 Set 1 | Question: 1

Question

Let $p, q, r, s$ represents the following propositions.

• $p:x\in\left\{8, 9, 10, 11, 12\right\}$
• $q:$ $x$ is a composite number.
• $r:$ $x$ is a perfect square.
• $s:$ $x$ is a prime number.

The integer $x\geq2$ which satisfies $\neg\left(\left(p\Rightarrow q\right) \wedge \left(\neg r \vee \neg s\right)\right)$ is ____________.

Answer 1

$\neg((p → q) \wedge (\neg r \vee \neg s))$
$\quad \equiv (\neg(\neg p \vee q)) \vee (\neg(\neg r \vee \neg s))$
$\quad \equiv (p \wedge \neg q) \vee (r \wedge s)$

which can be read as $(x\in  \{8,9,10,11,12\}$ AND $x$ is not a composite number$)$ OR $(x$ is a perfect square AND $x$ is a prime number$)$
Now for

• $x$ is a perfect square and $x$ is a prime number, this can never be true as every square has at least $3$ factors, $1,x$ and $x^2.$

So, second condition can never be true.
which implies the first condition must be true.
$x\in \{8,9,10,11,12\}$ AND $x$ is not a composite number

But here only 11 is not a composite number. so only $11$ satisfies the above statement. 

ANSWER $11.$

Comments

answer 1 thing,

satisfy means either tautology or contingency then why only choosing number to prove it as TRUE?

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Actually each of $p, q, r, s$ are predicates with single variable $x$. Since expression given is not quantified so have to replace $x$ with a value to turn the predicate into a proposition.
The questions asks for a $x$ which satisfy the given expression.

Answer 2

~((p → q) ⋀ (~r ⋁ ~s)) 
= (~(~p ⋁ q)) ⋁ (~(~r ⋁ ~s)) 
=(p ⋀ ~q) ⋁ (r ⋀ s)

now all whole numbers > 1 are either prime or composite . 

taking that into consideration..  ~q is equal to s

so equation becomes (p ⋀ ~q) ⋁ (r ⋀ s) = (p ⋀ s) ⋁ (r ⋀ s) = ( p ⋁ r ) ⋀ s 

now to above expression to become tautology .. s should be true...

only prime number in our set is 11...

So answer would be 11

Answer 3

(p ⇒ q) will give {8, 9, 10, 12} ¬r will give {8, 10, 11, 12} ¬s will give {8, 9, 10, 12} (¬r ∨ ¬s) will give {8, 9, 10, 11, 12} (p ⇒ q) ∧ (¬r ∨ ¬s) will give {8, 9, 10, 12} ¬((p ⇒ q) ∧ (¬r ∨ ¬s)) will give 11. Thus, C is the correct option.

Comments

this seems to always work. I'm still hesitant. can you explain why it always works.?

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arush_verma

we can replace ^ with Ո and ˅ with U.

Answer 4

Answer 11

Comments

explain plz

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First simplify the equation... then take any number from that set and try to get true value of the equation by applying the number...  u can see that 11 is satisfying the equation ...