Discrete Math Practice Question 2: (the practice quiz grades All of the Above as

Question

Discrete Math Practice Question 2:

(the practice quiz grades All of the Above as incorrect, and also, only one answer is correct)

To prove that "An integer n is divisible by 7 if and only if n2 is divisible by 7," we need to prove two separate implications. The answer choices list pairs of implications. For which pair(s) would proving both statements constitute a proof of the given biconditional statement? If n is an integer such that nis divisible by 7, then n2 is divisible by 7. If n is an integer such that n2 is not divisible by 7, then n is not divisible by 7 0 If n is an integer such that n is not divisible by 7, then n2 is not divisible by 7. If n is an integer such that n is not divisible by 7, then n is not divisible by 7. O If n is an integer such that n is divisible by 7, then n2 is divisible by 7 If n is an integer such that n is not divisible by 7, then n2 is not divisible by 7. If n is an integer such that n is not divisible by 7, then n2 is not divisible by 7. If n is an integer such that n2 is divisible by 7, then n is divisible by 7. O All of the above.

Explanation / Answer

Let us consider P=n is divisible by 7

Q=n2 is divisinbleby 7

So here we want to prove two cases first one is, "if n is divisible by 7 then n2 is divisible by 7 " and seond one is "if n2  is divisible by 7 then n is divisible by 7" . That is P imples Q and Q implies P.

But we know that P implies Q and -Q implies -P are equivalent.so we can rewrite this as "if n2 is not divisibe by 7 then n is not divisible by 7" and "if n is not divisible by 7 then n2 is not divisible 7".

So second one is correct.

First one is wrong since it does not prove Q implies P.

Third one also true since it does not prove,both P implies Q and -P implies -Q(Q implies P).

Fourth one false.since it does not prove P implies Q.

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