Studying for a test, help would be appreciated. Suppose you wish to prove P righ

Question

Studying for a test, help would be appreciated.

Suppose you wish to prove P rightarrow Q using the contrapositive method. Suppose P and deduce Q. Suppose ~ P and deduce ~ Q. Suppose ~ Q and deduce ~P. Suppose ~ P and ~Q. Suppose you are trying to prove the following Proposition by using direct proof: Proposition 1: If n is a positive integer such that n 2 mod 4 or n 3 mod 4, then n is not a perfect square Which of the following should be the first sentence of your proof. Suppose n is a positive integer such that n 2 mod 4 or n 3 mod 4. Suppose n is a positive integer such that n 2 mod 4 and n 3 mod 4 Suppose n is not a perfect square. Suppose n is a perfect square.

Explanation / Answer

1. (C)

Suppose ~Q and ~P

Because to show P implies Q we need to show not Q implies P.

2. (D)

3. (D)

For contrapositive we start with negation.

4. (A)

Because eventually you have to finish at the required result.

5. (C)

6. (B)

Suppose x^5-4x^4+3x^3-x^2+3x-2 > 0 then x>0 which is contradicting.

7. (B)

Same as above

8. (B,C,D)

All follow congruence rule

a = b mod (c)

Implies a-b is divisible by c

9. (A)

Because 14 divides a-b hence multiples of 14 also divide a-b

10. (B)

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