Cases, Counterexamples, and Contradictions

Videos (~40 mins)

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

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

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

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 particular, you might find it useful to use anything previously proven in a video – just cite the fact and move on, without proving it again.

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

Hint: Consider whether \(n\) is odd or even.