Question about implications

Are my implications here correct?

x = a
=> x^2 = a^2
<=> 2x^2 = 2a^2
=> x =a or x = -a

Are those ok? I'm just not sure if that <=> should be in the middle there. I mean, the implication goes both ways if we look at that line and the previous, but it doesn't go both ways if we look at all the lines before it.

Reply 1

SimonM

18

Yes

Consider each statement separately.

Reply 2

Swayum

18

Is it wrong to say x^2 = a^2 ==> x = a? The only time P ==> Q fails is when P is true but Q is false. But Q here is x = a, which can be true even though it's not necessarily true. Do we need Q to be neccesarily true in order to write the implication sign? Surely if P was false, Q being false wouldn't matter - how is that taken into account when writing implications in general in mathematics?

Also, why doesn't (x = a or x = - a) ==> 2x^2 = 2a^2?

Reply 3

Kolya

17

Swayum

Is it wrong to say x^2 = a^2 ==> x = a?

Yes.

Swayum

The only time P ==> Q fails is when P is true but Q is false. But Q here is x = a, which can be true even though it's not necessarily true. Do we need Q to be neccesarily true in order to write the implication sign? Surely if P was false, Q being false wouldn't matter - how is that taken into account when writing implications in general in mathematics?

We don't need Q to be necessarily true. I think the misunderstanding lies in the fact that your definition of => is slightly incomplete. You should say that "P -> Q" just in case in all the possible worlds in which P is true, Q is also true. In other words, P->Q doesn't hold if there exists a POSSIBLE WORLD in which P is true and Q is false. (And in the case under consideration, there does exist such a world. The world where x = -2 and a = 2 makes P true, but Q false.)

This "possible worlds" view should hopefully make things clearer to you. It is how you treat cases where the truth valuation of P and Q depends on the world you are looking at.

Also, why doesn't (x = a or x = - a) ==> 2x^2 = 2a^2?

It does.

Reply 4

birdsong1

Swayum

Is it wrong to say x^2 = a^2 ==> x = a? The only time P ==> Q fails is when P is true but Q is false. But Q here is x = a, which can be true even though it's not necessarily true. Do we need Q to be neccesarily true in order to write the implication sign? Surely if P was false, Q being false wouldn't matter - how is that taken into account when writing implications in general in mathematics?

Also, why doesn't (x = a or x = - a) ==> 2x^2 = 2a^2?

I think an easier way to think about it is just to read it in english. x^2 = a^2 ==> x = a reads "x^2 = a^2 implies x = a".

But clearly that's not the case, because you could have x = 1, a = -1, and then you'd have x^2 = a^2 but not x = a. So it can't be the case that x^2 = a^2 implies x = a.