Submitted by mrmonkey on Wed, 08/30/2017 - 15:11

I'm struggling to understand the difference between "if", "if and only if" and "only if"

to my understanding :

"A if B" means B⟹A.

"A if and only if B" means A⟺B so A implies B and B implies A.

Both of these make sense.

A only if B means "only if B is true, A is true", so B⟹A. But if A is true, as B being true is the only statement that makes A true, B must be true?...so A⟹B as well? which is the same as "if and only if"?

Can someone give some statements that are clear to explain the differences to me, especially a statement for "only if" that is true.

Hope this makes sense tia

## reading more about it, i

reading more about it, i understand that A only if B means A implies B? Is that all it means

## This is mostly correct. A if

This is mostly correct. A if B means that if B is true then A is true. Or equivalently $B \implies A$.

A only if B means that only if B is true is A then true. All this says is that it is impossible for $A$ to be true without $B$ being true as well. So if you know that $A$ is true, it must be the case that $B$ has to be true as well, so $A \implies B$. However it says nothing about $B$ being true implying that $A$ is true. So $B \not\imply A$. So A only if B means $A \implies B$ only.

$A \iff B$ or A if and only if B means that $A \implies B$ and $B \implies A$, so if one of $A$ or $B$ is true, the other must be as well.

Some examples:

If $B = (x = 2)$ then $A = (x^2 - 2x = 0)$. That is $B \implies A$. So $A$ if $B$. But it is not the case that $x^2 -2x = 0 \implies x=2$, since if all you know is that $x^2 - 2x = 0$, you cannot conclude that $x=2$. It could be that $x=0$.

## thanks

for the explanation, i understand now, A only if B means that B must be true for A to be true, but there may also be other conditions for A to be true. Ie if A is true you can conclude that B is true, but if B is true you can't necessarily conclude that A is true.

## Yep, although I have a minor

Yep, although I have a minor quibble with "there may be other conditions for $A$ to be true" this is sort of implying that $B$ is one possible condition that guarantees the truth of $A$, which is not the case. Essentially, $A$ only if $B$ says that it is impossible for $A$ to be true without $B$ being true, but this is not quite the same thing as saying that $B$ is true is a condition that makes $A$ true. You seem to have gotten the general grasp of it, though.