4. the reasoning is/or is not a proof of the statement same goes for 5,6,7 In th
ID: 3027761 • Letter: 4
Question
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.Explanation / Answer
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
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