Relations and equivalence relation

Question

Let $R=\{ (x,y) \vert x=1 \,\, or\,\, y=1 \}$

When I see something written like this to represent "or", I immediately think XOR. But is that necessarily true? This would greatly change the relation if only one variable can be 1.

Answer 1

In mathematics and in logic, you need to train your mind to automatically assume the inclusive-or, and understand that the exclusive sense of "or" holds if and only if the statement explicitly rules out the possibility that both disjuncts are true.

So your relation, $$R=\{ (x,y) \vert x=1 \,\, \text{or}\,\, y=1 \} = \{(x, y) \mid x = 1 \lor y = 1\}$$ means that one or both members of the ordered pair $(x, y)$ is equal to one. So indeed, $(1, 1) \in R$.

ADDED: You are indeed correct that if $R^*$ were defined so that $$R^* = \{(x, y)\mid x=1 \oplus y = 1\},$$ then it would change the relation considerably:

Specifically, we would have that $R^*$ fails to be reflexive and fails to be transitive, and hence, would not be an equivalence relation, whereas $R$, with the inclusive-or, is indeed an equivalence relation.

$R^*$ fails to be transitive because $(1, 0) \land (0, 1) \in R^*,$ but $(1, 1)\notin R^*$. And since $(1, 1) \notin R^*$, it fails to be reflexive.