## Proofs

Prove that for all integers m and n, m+n and m-n are either both odd or both even.

• ill assume that's inclusive or

we're given that m and n are integers and that mn is even...

suppose m is odd and n is odd
then m=2a+1 and n=2b+1 where a and b are integers
then mn=4ab+2a+2b+1
if we let c=2ab, then c is an integer
mn=2c+2a+2b+1
mn=2(a+b+c)+1
let d=a+b+c, then d is an integer
mn=2d+1
2d is an even integer by definition, and 2d+1 is an odd integer
but we started with the assumption that mn is even. this is a contradiction, therefore either
1. m is odd and n is even
2. m is even and n is odd
3. both m and n are even
and by definition, either m is even or n is even.
• Anonymous commented

That's not what the question asked

• if m is odd and n is odd, then m+n is even and m-n is even

if m is odd and n is even then m+n is odd and m-n is odd

and vicecersa

so both are either even or odd
• If m and n are both odd, then m = 2a + 1, n = 2b + 1 for some integers a and b.

So mn = (2a + 1) (2b + 1) = 4ab + 2a + 2b + 1 = 2 (2ab + a + b) + 1 is odd.

The proof above uses the method of contrapositives, i.e. the observation that the implication p ? q is logically equivalent to its contrapositive ¬q ? ¬p, so (dis)proving one (dis)proves the other automatically.
• if one is odd and one is even we have odd number when adding or subtracting
similarly if both are even we have even number when adding or subtracting
similarly if both are odd we have even number when adding or subtracting
• ##### Question Details
Prove that for all integers m and n, m+n and m-n are either both odd or both even.
ALL SYMBOLS USED ARE INTEGERS
CASE 1
M AND N ARE EVEN ..
M=2P.....N=2Q
M+N=2P+2Q=2[P+Q]=2* INTEGER = EVEN ...
M-N=2P-2Q=2[P-Q]=2*INTEGER = EVEN .....
M+N AND M-N ARE BOTH EVEN ....OK

CASE 2

M AND N ARE ODD

M=2P+1 .....N=2Q+1

M+N=2P+1+2Q+1=2[P+Q+1]=2*INTEGER =EVEN ....

M-N = 2P+1-2Q-1=2[P-Q]=2*INTEGER =EVEN .....

M+N AND M-N ARE BOTH EVEN ....OK

CASE 3

ONE OF THEM IS EVEN AND THE OTHER IS ODD ..SAY

M=2P+1.....N=2Q

M+N=2P+1+2Q=2[P+Q]+1=2*INTEGER + 1 = ODD

M-N=2P+1-2Q=2[P-Q]-1=2*INTEGER -1 =ODD

M+N AND M-N ARE BOTH ODD ....OK

HENCE THE RESULT

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