Why don't we use xor more often in ordinary mathematics?
For example, every integer is even xor odd; for every real number $x\ne y$, we have $x<y$ xor $x>y$; a graph is bipartite xor it contains an odd cycle; given an ultrafilter in a boolean algebra, every element xor its complement in the boolean algebra is in the ultrafilter; etc.
My main question is about the notation xor used to label the connective, but a related question would be on why the logical connective xor itself doesn't appear more often in our proofs and theorems. My broader question therefore would be about why we are inclined to use some logical connectives more often in mathematics and others not so much (eg. nand). Is there a meta-mathematical reason for this?
The xor of two statements is the equivalence of one of them to the negation of the other. More generally, the xor of $n$ statements is defined as the assertion that the number of them that are true is odd. This is especially inconvenient in stating either the antecedents or consequents of theorems, which are naturally stated as material conditionals and hence as inclusive disjunctions. And we frequently wish to state some $n$ statements are all equivalent. Further, the inclusive disjunction of $n$ statements - i.e. the claim that at least one of them is true - is also frequently useful, e.g. because some claim of interest follows from any one of them.