# Is $a \iff b$ a contradiction if when $a=T$ then $b=F$ and when $b=F$ then $a=T$?

Let $a,b$ be logic propositions. In the truth table for $a,b$ whenever $a=T$ then $b=F$ and when $b=F$ then $a=T$. Which of the below statements is true:

1. $a\to b$ is a tautology

2. $a\to b$ is a contradiction

3. $a \lor b$ is a contradiction

4. $a \iff b$ is a contradiction

5. none of the statements are correct

I think that 4) is correct that is $a \iff b$ is a contradiction because of the given that $a$ and $b$ are always the opposite of each other and $\iff$ requires both sides to be true.

But I'm not sure.

• Yes: $(a \leftrightarrow \lnot b)$ is equiv to $\lnot (a \leftrightarrow b)$. – Mauro ALLEGRANZA Jun 12 '18 at 15:38
• If you write the statements with logic symbols, maybe it's clearer: the statements you made are that $(a \rightarrow \neg b) \wedge (\neg b \rightarrow a)$, which means that... EDIT: exactly what @MauroALLEGRANZA wrote :) – Riccardo Sven Risuleo Jun 12 '18 at 15:38
• You wrote "⟺ requires both sides to be true" which is false. Equivalence does not mean both sides must be true. – user21820 Jun 16 '18 at 8:11

2) is true, since $a \to b$ holds iff $a = F$ or $b = T$, neither of which is the case.
3) is false, since $a \vee b$ holds iff $a = T$ or $b = T$.
4) is true, since $a \to b$ is a contradiction, as just mentioned. But $a \leftrightarrow b$ is an even stronger statement, so it is also a contradiction.