Cases, Contradictions, and Counterexamples

Videos (~30 minutes)

Give a counterexample to each of the following claims. Describe in a sentence or two why your counterexample shows that the claim is false.

Note: If the claim refers to multiple numbers like \(x\) and \(y\), a complete counterexample includes values for both \(x\) and \(y\).

\(\forall x,y \in \mathbb{Z} \;:\; x^2 < y^2 \rightarrow x < y\).
- Recall that \(\forall x,y\in \mathbb{Z}\) is a notation shortcut for \(\forall x \in \mathbb{Z}, \forall y \in \mathbb{Z}\).
\(\lnot \exists x \in \mathbb{Z}\;:\; x | 51\)
- Recall that \(a|b\) means that \(a\) is a divisor of \(b\) with remainder \(0\).
\(\forall x, y \in \mathbb{Z} \;:\; x^2 + y^2 \text{ is an even number}\).

Prove that \(\sqrt{6}\) is not a rational number; i.e. there are no integers \(a,b \in \mathbb{Z}\) such that \(\sqrt{6} = \frac{a}{b}\).

Notes:

A fraction \(\frac{a}{b}\) is in lowest terms if there is no integer \(k\) that divides both \(a\) and \(b\).
It is traditional in math classes that you can use anything you have previously proven. You might find a problem from the last warmup to be helpful.
It is also traditional that you may “use without proof” facts that the instructor tells you are acceptable to use without proof. In this case, you may use without proof the fact that, for any integer \(n\), it is the case that \(n^2\) is even if and only if \(n\) is even.

Prove that if \(n\) is an integer, then \(3n^2 + n + 4\) is even.

Hint: Divide into cases based on whether \(n\) is odd or even.