Worksheet Practicing Writing Proofs
Activity 1.2.
Prove the theorem “For every integer \(x\text{,}\) \(x+4\) is odd if and only if \(x+7\) is even.” (Recall that this requires proving two separate implications. Identify them before writing your proofs!)
Another type of theorem that we are sometimes called upon to prove is a uniqueness theorem. In such a case, we must prove that the thing in question exists as well as prove that there is only one such thing. The next activity walks you through the structure of such an argument.
Activity 1.3.
We want to prove the theorem “If \(c,a,r\in\mathbb{R}\) such that \(c\neq 0\) and \(r\neq a/c\text{,}\) then there exists a unique \(x\in\mathbb{R}\) such that \((ax+1)/(cx) =
r\text{.}\)”
(a)
The first step in a uniqueness proof is to demonstrate that such a beast exists. One pattern for doing this is by coming up with a formula for the thing (here \(x\)) that we need to have exist and showing that \(x\) does what it’s supposed to. The process of how you come with the \(x\) is scratchwork that doesn’t go in the proof!. Thus, we want to start by finding a formula for \(x\) so that \((ax+1)/(cx) = r\text{.}\) Use familiar algebra skills to do it. What you write up for your proof is going to look like “Let \(x=\cdots\text{.}\) Now substituting and simplifying, we see:” starting with \((ax+1)/(cx)\) and simplifying to get \(r\text{.}\)
(b)
Next, we have to show that our \(x\) is unique. To do so, you assume that \(x_1,x_2\in\mathbb{R}\) both have the property required. Namely, that \((ax_1+1)/(cx_1) =
r\) and \((ax_2+1)/(cx_2) =
r\text{.}\) Now you use algebra to show that \(x_1=x_2\text{,}\) which means that the \(x\) you came up with originally really is unique.
(c)
What you have above likely has a lot of scratchwork in it. Rewrite things neatly to have a nice paragraph that proves the theorem.
Activity 1.4.
Pick and choose amongst the following for more practice.
(a)
If \(n\) is even, then \(n^2\) is even.
(b)
If \(n\) and \(m\) are even, then \(nm\) is even. (What do you notice about your proof?)
(c)
Prove or disprove: If \(n\) is odd, then there exists an integer \(m\) such that \(n^2=8m+1\text{.}\) (If you decide the statement is true and prove it, can you prove that \(m\) is unique?)
(d)
For every integer \(n\text{,}\) if \(n^2\) is an odd integer, then \(n\) is an odd integer.
