Negation of exclusive or

Question

Suppose $A$ and $B$ are two statements.

What is the negation of the excluisive or-statement, i.e. of "either $A$ or $B$" which i formally written as $A\dot{\vee}B$?

I think $\neg (A\dot{\vee} B)$ means

($A$ and $B$) or (not A and not B), i.e.

$$ \neg(A\dot{\vee} B)=(A\wedge B)~\vee~(\neg A\wedge\neg B) $$

(the or on the LHS is exclusive while the or on the RHS is inclusive).

Which is equivalent to the "if and only if" proposition $A \leftrightarrow B$. — Daniel Schepler, Sep 25 '18 at 18:23

score 2 · Answer 1 · answered Sep 25 '18 at 18:15

2

That is correct. An equivalent (by DeMorgan's laws) statement is $$(A \vee \lnot B) \wedge (\lnot A \vee B)$$

answered Sep 25 '18 at 18:15

Ross Millikan

374,822

score 0 · Answer 2 · answered Sep 26 '18 at 19:18

You are going to get confused if you use the same symbol for inclusive or and exclusive or. $\lor$ is almost invariably used for inclusive or, so I suggest you use something else (maybe $\oplus$) for exclusive or. In classical logic, the exclusive or $A \oplus B$ holds precisely when $A$ and $B$ have different truth values (do you see why?), so $\lnot(A \oplus B)$ holds precisely when $A$ and $B$ have the same truth value, i.e., when $A \Leftrightarrow B$, or equivalently $(A \land B) \lor (\lnot A \land \lnot B)$, holds. So the right-hand side of your equation is right, but I would advise you to use different notation on the left-hand side.

Negation of exclusive or

2 Answers2