Solving a version of the liar paradox

Question

Given two people $Alice ,Bob$ are either lying or telling the truth Now suppose $Alice$ says "At least one of us is lying."

We have two cases:
$Alice$ is telling the truth $\implies$ $Bob$ is lying.
$Alice$ is lying, $Alice$ is lying about $\exists$ liar, thus $Bob$ must be telling the truth. Now this is a familiar form of the liar paradox, and for homework I'm given 5 answer choices:
(1) Alice and Bob are either both liars or both say the truth.
(2) Alice and Bob are both liars
(3) Alice and Bob both say the truth
(4) Alice says the truth, Bob is a liar
(5) Alice is a liar, Bob says the truth

Any help is appreciated

Answer 1

The principle here is that

If x says y then $ x \equiv y $ is true independent of if x is telling the truth or is lying.

So in your question allice says "one of us is an liar"

"one of us is an liar" can be formulated as $ \lnot a \lor \lnot b $

The formula to test then becomes $ a \equiv (\lnot a \lor \lnot b) $ at that you can do with a truth table.

n | a | b | a <-> ( ~a v ~b)
1 | T | T |    F       F
2 | T | F |    T       T
3 | F | T |    F       T
4 | F | F |    F       T

Only in line 2 the formula is true so $Alice$ is telling the truth and $Bob$ is a liar.