4 the reasoning isor is not a proof of the statement same go

4. the reasoning is/or is not a proof of the statement

same goes for 5,6,7

In this question you are asked to evaluate a reasoning proposed as a proof a given statement. You are also asked to decide if the statement is true. Statement: If 7p + 8 is odd then p is odd. Reasoning: By contrapositive, we assume that 7p + 8 is even (negation of the premise) and aim at showing that p must be even (negation of the conclusion). If 7p + 8 is even then 7p + 8 = 2k for some integer k. Therefore 7p = 2k - 8 = 2(k - 4). Since k - 4 is an integer, it follows that 7p is divisible by 2 (i.e., is even). But 7 is not divisible by 2; so p must be divisible by 2, i.e., p must be even. This completes the proof by contrapositive of the statement. he Reasoning $ a proof of the Statement. The Reasoning a proof on the converse of the Statement. The Reasoning a proof by contra positive of the Statement. The Reasoning proof of the converse of the Statement. The Statement true.

Solution

This proof by contraposition and proof by contradiction are very similar. For proof by contraposition of a statement p->q, assume ~q and show ~p. For proof by contradiction assume ~(p->q), which is equivalent to assuming (p ^ ~q), and show that it leads to a contradiction.

if 7p+8 is odd, then p is odd using a

a) proof by contraposition
Prove the statement by proving the contrapositive. Assume p is even and show that 7p + 8 is even.
p = 2k for some integer k by the definition of even integers.
7p +8 = 7(2k) + 8

= 14k +8

=2(7k+4) , so 7p+4 is 2 times an integer , so it is even.

b) proof by contradiction:
Show that if the statement is F, it leads to a contradiction. The statement is F when 7p + 8 is odd, and n is even, so assume 7p + 8 is odd, and p is even.

p = 2k for some integer k by the definition of even integers. 7p +8 = 7(2k) + 8

= 14k +8

=2(7k+4) , so 7p+8 is even, but we assumed 7p+8 was even. The

contradiction completes the proof.

in the above reasoning is proof of the statement.if 7p+8 is odd then absolutely then p is also odd