GO Classes CS Test Series 2025 | Discrete Mathematics | Topic Wise Test 1 | Question: 5

Answer 1

We can ONLY infer that $7$ is antisocial.

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Answer 2

Domain: Set of natural numbers.

Let,

  P(n): n is prime
  A(n): n is antisocial

Then,

If n is prime, then n in antisocial $ \equiv \forall_n(P(n) \to A(n))$

Now considering n = 10, and we know $P(10) \equiv False$

  $P(10) \to A(10)$                   $ \because $ Universal Instanstiation.
  $\neg{A(10)} \to \neg{P(10)}$            $\because A \to B \equiv \neg{B} \to \neg{A}$
  $\neg{P(10)}$
$--------$
Cannot infer anything

Now considering n = 7, and we know $P(7) \equiv True$

  $P(7) \to A(7)$                        $ \because $ Universal Instanstiation.
  P(7)
$--------$
$\therefore A(7)$

$\therefore$ Correct option is C. 7 is antisocial

Comments

I am not clear on how to go about inferring/arriving at a conclusion when premises are given.

Like here in the answer.. in the first argument, how do I frame that premise, and how to arrive at the conclusion/inference? @d0t can you elaborate more on that part please, and let me know what key concept am I missing here.

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@Shubhamishere I think the concept your missing here is “Rules of inferences”, there are some simple rules by which we can infer or arrive at the conclusion. For eg in this question it is given to us that 

Let S be “If n is prime, then n in antisocial” which is equivalent to $ \forall_n(P(n) \to A(n))$. Domain is Set of all Natural numbers, by this we can say that for all set of natural numbers S should be true, then we can also say that for a particular value n, when n = 10, S should also be true i.e  $P(10) \to A(10)$ 

Next we are given that 10 is not a prime number and 7 is a prime number i.e P(10) is false and P(7) is true

We know that  $P(10) \to A(10)$ is true and $\neg{P(10)}$ is true but with this we cannot infer anything about whether A(10) is True or False because “F → Anything” is anyway True.

but in the second case we know P(7) is true and $P(7) \to A(7)$ is true we were able to infer that A(7) has to be true because then this $P(7) \to A(7)$ statement would be false otherwise that case is (T → F). 

Here the premises are itself given in the question. For further reading you can either go through Kenneth Rosen Chapter 1.6 Rules of inference or you can watch this lecture and the next one in the sequence. 

https://www.goclasses.in/s/courses/63f9aa9be4b0a8a370cfb0ad/take

 

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thanks for the detailed explanation, went through that topic. it helped me clear my concept gap.

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Answer 3

According to the question : If n is prime then n is antisocial.

prime -> antisocial

a. 10 is not prime. (Not Correct)

b. prime -> antisocial means ~antisocial-> ~prime   So we cant say 10 is antisocial

c. if 7 is prime then 7 is antisocial it straight forword option which is correct.

d. Same logic of option b.

 

Only C is correct.

#Jai_Hind