Formula F is equivalent to formula G iff
A) F IFF G is valid
B) F IFF NOT(G) is not valid
C) F XOR G satisfiable
D) F XOR G is not satisfiable
I have been told solution D is correct.
How is this statement correct?
Statement: Formula F is equivalent to formula G iff F XOR G is not satisfiable.
I know that if two formulas are equivalent then they are both valid(F<-->G) . However, I don't understand how F XOR G not being satisfiable would apply here. When you XOR, the result true whenever an odd number of the inputs are true and is false whenever an even number of inputs are true. So it could never be valid or never "not satisfiable", right?
See XOR Truth table.
To say that $F \text { XOR } G$ is not satisfiable means that it is always FALSE.
But, due to the corresponding truth table, the formula is always FALSE only when either $F$ and $G$ are both TRUE or they are both FALSE.
And two frmulas $F$ and $G$ are equivalent when either they are both TRUE or both FALSE (alternatively, when $F \Leftrightarrow G$ is valid or tautological).