Suppose a and b are positive integers. Write a proof by contradiction to show that if a-b is odd,then a+b is odd.

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To prove by contradiction that if a-b is odd, then a+b is odd, we assume the opposite: that if a-b is odd, then a+b is not odd.

So, let's assume that a-b is odd, but a+b is even.

Since a-b is odd, we can express it as (2k+1), where k is an integer. This is the definition of an odd number.

Now, let's express a+b as 2m, where m is an integer. This is the definition of an even number.

Now, based on the assumption, we have:

a-b = 2k+1

a+b = 2m

Adding both equations:

(a-b) + (a+b) = (2k+1) + 2m

Simplifying the equation:

2a = 2k + 1 + 2m

Now, rearranging the terms:

2a = 2(k + m) + 1

This shows that 2a is odd since it has the form (2n + 1), where n is an integer.

However, this contradicts the assumption that a+b is even. Since we arrived at a contradiction, our initial assumption must be false.

Therefore, if a-b is odd, then a+b must also be odd.

That concludes the proof by contradiction.