Why don't we use xor (or nand)? 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?
 A: 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.
A: One reason is that it is easier to prove a non-exclusive disjunction: simply prove one disjunct or the other.
Another reason is that it is easier to define the exclusive disjunction in terms of the non-exclusive disjunction than the other way around: the exclusive disjunction is the non-exclusive disjunction plus the claim that the disjuncts are mutually exclusive. If you are writing a proof, you can simply say something like "either $A$ or $B$, but not both" to express the exclusive disjunction. There is no reason to introduce the neologism "xor" into the language when a simple "but not both" will do.
