Say we're trying to prove by contradiction that if n2 is an odd number, then n is also odd for all integers n. What's our proposition? What's our supposition?
Hint
One proposes, and the other supposes.
Answer
Our proposition is: "If n2 is an odd number, then n is also odd for all integers n." Our supposition is the opposite: "There exists an integer n such that n2 is odd and n is even."
Example 2
If our supposition in a proof by contradiction was "there exists some integer n such that the product of n and its reciprocal does not equal 1," what was our proposition?
Hint
The proposition was the original statement we were trying to prove.
Answer
"The product of an integer and its reciprocal always equals 1."
Example 3
Prove the following statement by contradiction:
There is no integer solution to the equation x2 – 5 = 0.
Hint
Our Opposite-Day supposition is "there is at least one integer solution to the equation x2 – 5 = 0."
Answer
Since x2 – 5 = 0 implies that x2 = 5, and there aren't any integers we can square to get 5, this contradicts the supposition that there is at least one integer solution to the equation x2 – 5 = 0. Hence, our original proposition was true: there is no integer solution to the equation x2 – 5 = 0.