# Understanding the simbolization of an “either or” in proofs.

In the book "How to Prove It", by Velleman, appear these two examples.

1. Prove that for every integer $$x$$, the remainder when $$x^2$$ is divided by 4 is either 0 or 1.
2. Prove that for every real number x, if $$x^2 \geq x$$ then either $$x \leq 0 \lor x \geq 1$$.

Symbolization:

1. $$x \in \mathbb{Z} \to (x^2 \text{ has remainder } 0) \lor (x^2 \text{ has remainder } 1)$$
2. $$\forall x(x^2 \geq x \to (x \leq 0 \lor x \geq 1)$$

In both cases, the author symbolises the conclusion with a disjunction.

In my mind, the conclusion is an exclusive or. In the first example, if I write a number in the form $$k \in \mathbb{Z}$$, that same number can not be written in the form $$4l$$ for some number $$l \in \mathbb{Z}$$. In the second one, a number being 0 excludes the possiblity of it being greater than or equal to 1.

What would be the explanation for this issue, from a logic perspective ?

If my perspective is incorrect, how does a proof of an "exclusive or" look like ?

• I think this covers the main aspects of the question. – Gae. S. Apr 25 '20 at 21:17
• Why in the symbolization of 1. appears $\mathbb{R}$ instead of $\mathbb{Z}$? – Riccardo Apr 25 '20 at 21:35

Since in these cases the options are mutually exclusive, both kinds of or are equivalent. It's usually more convenient in mathematics to use $$\lor$$ if possible, because it's dual to $$\land$$, and many proofs easily obtain a result of the form $$p\lor q$$, where $$p\oplus q$$ is either harder to prove or in general false. For example, $$xy=0\to x=0\lor y=0$$ (unless we have zero divisors).
• Could you expand a little bit on your example $xy=0\to x=0\lor y=0$ ? – F. Zer Apr 25 '20 at 22:22
• @F.Zer The product of nonzero quantities is nonzero, so from $xy=0$ we can deduce $x=0\lor y=0$. But that's all we can deduce; we can't deduce $x=0\oplus y=0$. – J.G. Apr 25 '20 at 22:24