Tao Analysis I - A.1.2

Exercise A.1.2

What is the negation of the statement “X is true if and only if Y is true”? (There may be multiple ways to phrase this negation.)

This English statement can be interpreted in several ways if read informally. 

To a mathematician, the phrase "if and only if" has a precise meaning. It is the bi-conditional $X⟺Y$, which is the combination of two implications $(X⟹Y)\wedge (Y⟹X)$.

Let's try to negate the statement at an English level:

• "it is not the case that X is true if and only if Y is true"
• .. leads to "X is true if Y is false, and, Y is true if X is false"

It is not immediately apparent that this also means the statement is false when X and Y are both true, or both false. It needs some thinking. Due to this risk, it is good practice to write out the truth tables.

X	Y	X $⟹$ Y	Y $⟹$ X	X $⟺$ Y	$\mathrm{\neg }$ (X $⟺$ Y)
F	F	T	T	T	F
F	T	T	F	F	T
T	F	F	T	F	T
T	T	T	T	T	F

Here we've used the truth value definition of implication, which itself is counter-intuitive and requires a separate blog post to discuss.

The truth table confirms the nation of the given statement is "at most one of X or Y is true", or "either X or Y is true, but not both", or simply the exclusive-or relation $(X\oplus Y)$. $◻$

Note that combining this result and the one from the last exercise A.1.1, we can see XOR and logical equivalence are negations of each other.

$$\boxed{{\displaystyle \mathrm{\neg }(X\oplus Y)=(X⟺Y)}}$$