Semantics of exclusive or

Question

How do I get a description of $\text {XOR}$ (exclusive or) only using the operators $\wedge$, $\vee$, $\neg$, $\rightarrow$

And is it possible to prove the correctness of such description?

Answer 1

It is useless to look for correctness here.

" Exclusive OR" is simply the name that has been given to the following truth function , more precisely to the function from {T,F}² to {T,F} such that :

(T,T) --> F

(T,F) --> T

(F,T) --> T

(F,F) --> F

Look at this sagittal representation of the XOR function and ask yourself : what does it mean to say that (A XOR B) is true? If I say that : (A XOR B) , I say something true if and only if it is the case that : .....

If you compare the truth table of the X-OR operator to the truth table of the OR operator, you will see that X-OR is more demanding: in order A XOR B to be true, you need (1) what is required in order A OR B to be true, (2) plus an additional condition ...

So to find your translation, you only have to ask yourself: what does the X-OR operator add to the OR operator? In other words, what happens when an OR ( a disjunction) becomes EXCLUSIVE ( that is, is no longer inclusive)?