Question

X=2 if and only if |x| = 2
True or false? Please answer if you're 100 percent sure.

Answer 1

False

Explanation:

Let's just analyze it.

x=2 if and only if |x|=2

That can be rewritten as:

if x = 2 then |x| = 2

AND

if |x| = 2 then x = 2

Let's solve each one separately:

we have x = 2

we can simply get |x| by removing the sign, if any. There is none, so |x| = 2.

The first part checks out.

we have |x| = 2

As with absolute values, this means that either x = 2 or x = -2

So we know from |x| that x is either 2 or -2, but the logical statement said "if |x| = 2 then x = 2", which is too specific - it forgets about -2.

The second part is false.

Overall, we now know that "X=2 if and only if |x| = 2" is false because x = -2 is a counterexample. We get: "False if and only if True" which itself is false.