Homework SourseConsider the statement S If n2 is even then n is even Read t
Consider the statement:
S: If n^2 is even then n is even.
Read the texts A and B below carefully:
Text A:
Assume n^2 is even and n is odd.
Then n = 2 x k + 1 for some integer.
Then n^2 = (2 x k + 1)^2 = 4 x k^2 + 4 x k + 1 = 2 x (2 x k^2 + 2 x k) + 1
Therefore n^2 is an odd number.
Text B:
Assume n is odd.
Then n = 2 x k + 1 for some integer k.
Then n^2 = (2 x k + 1)^2 = 4 x k^2 + 4 x k + 1 = 2 x (2 x k^2 + 2 x k) + 1
Therefore n^2 is an odd number.
Text A: Choose the right answer:
a) is a fragment of a proof by contradiction of S
b) is a fragment of a proof by contrapositive of S
c) is a fragment of a direct proof of S
d) is not a fragment of a proof of S
Text B: Choose the right answer:
a) is a fragment of a proof by contrapositive of S
b) is a fragment of a proof by contradiction of S
c) is a fragment of a direct proof of S
d) is not a fragment of a proof of S
Consider further texts C and D:
Text C
This contradicts the assumption that n^2 is even.
Text D
We will prove that if n is odd then n^2 is also odd.
Text C: Choose the right answer:
a) fits with Text A
b) fits with Text B
c) fits neither with text A nor text B
Text D: Choose the right answer:
a) fits with Text A
b) fits with Text B
c) fits neither with text A nor text B
Solution
we have to prove that n is even iff n2 is even
but we have let that n is odd there we got n2 is odd
so if n is even then n2 is even and vice versa
we proved by contradiction
so the option is a) is a fragment of a proof by contradiction of S
