Let P be the set of prime numbers and let S = {t | 2t – 1 is a prime

1.	Let P be the set of prime numbers and let S = {t \| 2t – 1 is a prime}. Prove that S ⊂ P.
Answer» Now the equivalent contra positive statement of x ∈ S ⇒ x ∈ P is x ∉ P ⇒ x ∉ S. Now, we will prove the above contra positive statement by contradiction method Let x ∉ P ⇒ x is a composite number Let us now assume that x ∈ S ⇒ 2^x – 1 = m (where m is a prime number) ⇒ 2^x = m + 1 Which is not true for all composite number, say for x = 4 because 2⁴ = 16 which can not be equal to the sum of any prime number m and 1. Thus, we arrive at a contradiction ⇒ x ∉ S. Thus, when x ∉ P, we arrive at x ∉ S So S ⊂ P

Let P be the set of prime numbers and let S = {t | 2t – 1 is a prime}. Prove that S ⊂ P.

Answer»

Now the equivalent contra positive statement of x ∈ S ⇒ x ∈ P is x ∉ P ⇒ x ∉ S.

Now, we will prove the above contra positive statement by contradiction method

Let x ∉ P

⇒ x is a composite number

Let us now assume that x ∈ S

⇒ 2^x – 1 = m (where m is a prime number)

⇒ 2^x = m + 1

Which is not true for all composite number, say for x = 4 because

2⁴ = 16 which can not be equal to the sum of any prime number m and 1.

Thus, we arrive at a contradiction

⇒ x ∉ S.

Thus, when x ∉ P, we arrive at x ∉ S

So S ⊂ P