Question

Prove:
For all integers n, if n2 is odd, then n is odd.
Use a proof by contraposition, as in Lemma 1.1.
Let n be an integer. Suppose that n is even, i.e., n = ? for some integer k. Then n^2 = ? = 2( ? )
is also even.

Answer 1

that might be easier to see

Answer 2

contraP is to flip and negate the prepositions right?

if A then B, if -B then -A

Answer 3

ok, what does an even number look like?

Answer 4

2

Answer 5

that is a specific even number yes; but there is a general setup for an even number ...

for a given interger, |k| = 0,1,2,3,4,.... we can express an even number using k

Answer 6

think of your 2 times table, what does it represent?

Answer 7

so what needs to go inside the blanks

Answer 8

.... that is what we are figuring out, this is not a free answering service, it is a place to STUDY the material.

Answer 9

i understand i tried 2, 4, and 2 but it was incorrect

Answer 10

it was incorrect because you entered in a specific even value, and not a general expression that can define any even number

Answer 11

a proof requires generality, otherwise you are only proving it for a specific value and not ALL the values

Answer 12

2 times tables go like this:

2(0)=0
2(1)=2
2(2)=4
2(3)=6
2(4)=8
2(5)=10

would you agree that the 2 times tables ARE the even numbers?

Answer 13

yes

Answer 14

then in generality, 2, times an interger k, represents any even integer

so would you agree that 2k goes in the first box?

Answer 15

yes

Answer 16

now that we have an expression that is valid for ALL even integers, lets square it ..

what is (2k)^2 ?

Answer 17

4k^2

Answer 18

good, and we can write that as: 2(2k^2) , can 2k^2 be anything other than some integer?

Answer 19

so, second box looks to be 4k^2, and last box factors out the 2 like i did

Answer 20

yes that makes sense..u are a good teacher :)) thanks!

Answer 21

youre welcome, and good luck :)