As part of my discrete mathematics class, I introduce my freshmen/sophomore students to various proof techniques, including proofs about sets. Here is one of the examples that I use that involves adding and subtracting a number twice in the same proof.
Theorem. Let be the set of even integers, and define
Proof (with annotations). Before starting the proof, I should say that I expect my students to use the formal definitions of even and odd:
- An integer is even if for some integer .
- An integer is odd if for some integer .
To prove that , we must show that and . The first of these tends to trickiest for students.
Part 1. Let . By definition of even, that means that there is an integer so that .
To show that , we must show that for some odd integer . To this end, notice that . Thus, we must show that is an odd integer, or that can be written in the form . To do this, we add and subtract 1 a second time:
By the closure axioms, is an integer. Therefore, is an odd number by definition of odd, and hence $n \in B$.
The above part of the proof can be a bit much to swallow for students first learning about proofs. For completeness, let me also include Part 2 (which, in my experience, most students can produce without difficulty).
Part 2. Let , so that for some odd integer . By definition of odd, there is an integer so that $m = 2k+1$. Therefore, . By the closure axioms, is an integer. Therefore, is even by definition of even, and so we conclude that .
For what it’s worth, this is the review problems for which I recorded myself talking through the solution for the benefit of my students.
In my opinion, the biggest conceptual barriers in this proof are these steps from Part 1:
These steps are undeniably awkward. Back in high school algebra, students would get points taken off for making the expression more complicated instead of simplifying the answer. But this is the kind of jump that I need to train my students to do so that they can master this technique and be successful in their future math classes.