Prove that if x+y is odd, then x and y are of same parity.

[shivers20]

I am having some trouble constructing this proof.

Prove that if x+y is odd, then x and y are of same parity.

Proof: Assume x and y have same parity.

Case1: x and y are odd. x= 2k+1 , y=2m+1.

x+y= (2k+1) + (2m+1) = 2k+2m+2= 2(k+m)+1 I know I made a mistake here. How do I get rid of that =2?

Case 2: x and y are even. Foolow the same steps used in Case 1. Am I correct?

Now do I give a contrapositive?
"If x and y are two integers with opposite parity, then their sum must be odd." So we assume x and y have opposite parity. Since one of these integers is even and the other odd, there is no loss of generality to suppose x is even and y is odd. Thus, there are integers k and m for which x = 2k and y = 2m+1. Now then, we compute the sum x+y = 2k + 2m + 1 = 2(k+m) + 1, which is an odd integer by definition.

Have it done any of the proof correctly? Thankyou in advance.

 

[arthur ohlsten]

1 assume x and y are poth positive
then 2m + 2k =2[m+k] even pairty

2. assume x and y are both negative
2m+1 +2k+1=2[m+k]+2 even parity

3. assume x and y of opposite parity
2m+1 + 2k = 2[m+k]+1 odd parity

if x and y of same parity , sum is even parity
if x and y of different parity, sum is odd parity

 

[soroban]

Re: Prove that if x+y is odd, then x and y are of same parit

Hello, shivers20!

I am having some trouble constructing this proof . . . No wonder!

Prove that if \(\displaystyle x\,+\,y\) is odd, then \(\displaystyle x\) and \(\displaystyle y\) are of same parity . . . This is not true!

Please check the original wording of the problem.

 

[shivers20]

Re: Prove that if x+y is odd, then x and y are of same parit

Hello, shivers20!

I am having some trouble constructing this proof . . . No wonder!

Prove that if \(\displaystyle x\,+\,y\) is odd, then \(\displaystyle x\) and \(\displaystyle y\) are of same parity . . . This is not true!

Please check the original wording of the problem.

I cant believe I made that mistake. Sorry. Prove that if x+y is odd, then x and y are of opposite parity.

 

[soroban]

Re: Prove that if x+y is odd, then x and y are of same parit

Hello, shivers20!

Prove that if \(\displaystyle x\,+\,y\) is odd, then \(\displaystyle x\) and \(\displaystyle y\) are of opposite parity.

Assume \(\displaystyle x\) and \(\displaystyle y\) are of the same parity.
That is: [1] both are even, or [2] both are odd.

[1] Both are even.
. . Then \(\displaystyle x\:=\:2a,\;y\:=\:2b\) for integers \(\displaystyle a\) and \(\displaystyle b\).
Hence: \(\displaystyle \,x\,+\,y\:=\:2a\,+\,2b\:=\:2(a\,+\,b)\) ... a multiple of 2.
. . Their sum is an even number . . . a contradiction.

[2] Both are odd.
. . Then \(\displaystyle x\:=\:2a\,+\,1,\;y\:=\:2b\,+\,1\) for integers \(\displaystyle a\) and \(\displaystyle b\).
Hence: \(\displaystyle \,x\,+\,y\:=\2a\,+\,1)\,+\,(2b\,+\,1) \:=\:2a\,+\,2b\,+\,2\:=\:2(a\,+\,b\,+\,1)\) ... a multiple of 2.
. . Their sum is an even number . . . a contradiction.

Therefore, our assumption is incorrect: \(\displaystyle \,x\) and \(\displaystyle y\) are of opposite parity.