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.