# Symbol for “if and only if”: $\implies$ or $\iff$?

I was wondering about the iff sign in maths. I've never learned about it in school & see it a lot online. Usually the sign looks like this: $\implies$, but in math.stackexchange I always see this: $\iff$.

Does that sign means iff?

Am I using iff right here:

$$8x + 2x - 44 = 220 + 4 - x \implies 11x = 268 \implies x = 24.36$$

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No, $\implies$ is not $\iff$ . – Inceptio May 26 '13 at 15:35
==> is a simple implication – LeGrandDODOM May 26 '13 at 15:35
"$\iff$" $\implies$ "$\implies$", but "$\implies$" $\,\,\,\not\!\!\!\!\implies$ "$\iff$". – Willie Wong May 26 '13 at 16:59
@MetinY. Fortunately comments aren't answers. It's just a joke, not arrogance. – Thomas May 26 '13 at 21:50
@Thomas What Willie Wong says is not an entirely a joke. CEpqCpq is a theorem in classical propositional logic (the first part of what Willie Wong says), but CCpqEpq is not a theorem. His notation may look silly, but there do exist correct ideas behind it. – Doug Spoonwood May 29 '13 at 15:36

The expression $A \implies B$ means "if $A$ is true, then $B$ must be true". You can make a truth table: the truth or falsity of the statement $A\implies B$ is determined like this

$$\begin{array}{c|c|c|} & B\text{ is true}& B\text{ is false}\\\hline A\text{ is true} & \text{true} & \text{false}\\\hline A\text{ is false} & \text{true} & \text{true}\\\hline \end{array}$$

Similarly, $A\impliedby B$ means "if $B$ is true, then $A$ must be true". The truth table for $\impliedby$ is $$\begin{array}{c|c|c|} & B\text{ is true}& B\text{ is false}\\\hline A\text{ is true} & \text{true} & \text{true}\\\hline A\text{ is false} & \text{false} & \text{true}\\\hline \end{array}$$

The expression $A\iff B$ then means "both $A\implies B$ and $B\implies A$". For $\iff$, we get $$\begin{array}{c|c|c|} & B\text{ is true}& B\text{ is false}\\\hline A\text{ is true} & \text{true} & \text{false}\\\hline A\text{ is false} & \text{false} & \text{true}\\\hline \end{array}$$

Examples: for a real number $r$,

• $r>0\implies r^2>0$ (but $r>0\,\,\,\,\not\!\!\!\!\impliedby r^2>0$, because $(-1)^2>0$ even though $-1\not> 0$)

• $r=1\iff r+1=2$

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+100. I am fond of using tables. Nice Zev. – Babak S. May 26 '13 at 15:44
Quick question, how do you know that B or A is true? What makes it true or false? Sorry if you explained it already and I didnt get it hehe. – user73230 May 26 '13 at 15:44
@user73230: You may not know whether $A$ or $B$ is true; also, $A$ and $B$ may not be simple statements, but could have variables in them or whatnot. Think of $\Box\implies \Box$ as a function, that inputs two statements $A$ and $B$, and outputs true and false as the case may be according the rule specified. If I said "the function $f$ is defined by $f(x)=x^2$" or something like that, it is an incorrect question to ask "how do you know what $x$ is"; $x$ could be anything, that's the point. – Zev Chonoles May 26 '13 at 15:48
Similarly, $A\implies B$ is just a way of forming a new statement, whose truth value depends on the truth values of $A$ and $B$ (which can in turn depend on other things). – Zev Chonoles May 26 '13 at 15:50

The symbol $\implies$ means "implies" or "only if", and in $\LaTeX$, which you should use, it is called \implies.

The symbol $\iff$ means "if and only if (iff)" or "implies and is implied by" or "is equivalent to". It is \iff.

For example, $$x=1 \implies x^2=1$$ is a correct use but $$x=1\iff x^2=1$$ is incorrect because it is possible that $x=-1,0$ from the right hand side.

Your statement is true, but you could replace the $\implies$ with $\iff$ because they hold both ways.

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It is incorrect because we don't know how much is X right? so in that case: 2x + 1 = 4 - 1 ==> 2x = 2 ==> x = 1 <==> x^2 = 1 Is that correct? – user73230 May 26 '13 at 15:51
That's not correct when thought of as a bunch of pairwise relations because that expression contains, as a bit of it, $x=1\impliedby x^2=1$ which is not true. – Sharkos May 26 '13 at 15:53

To complete the answer to include what you mentioned in the title:

• "${}\Longleftarrow{}$" means "if",
• "$\implies$" means "only if", and
• "$\iff$" means "if and only if", which is sometimes written "iff".

A bare "if" not followed by "and only if" is actually rather rarely used. Instead of "$A$ only if $B$" it is common to say "$A$ implies $B$".

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Consider this:

"It's Christmas" implies "Current month is december"

But "Current month is december" does not imply "It's Christmas"

Both assertions are not equivalent, but "Christmas" implies the other

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As others have said, $A$ if and only if $B$ means that $A$ implies $B$ and $B$ implies $A$. That is, if $A$ is true then $B$ is true, and if $B$ is true then $A$ is true.

To make sense of why we say 'if and only if': If we have that $A$ if and only if $B$, then we have $A$ IF $B$ since $B$ implies $A$, i.e. IF $B$ is true, then $A$ is true.

We also have $A$ ONLY IF $B$, since we have $A \implies B$, and using basic logic (not sure if you know this - you would see it in an introductory abstract math course) we see that the negation of this is $\neg B \implies \neg A$, or in other words if $B$ is NOT true, then $A$ is not true. Therefore, $A$ is only true if $B$ is true, or $A$ only if $B$.

So combining $B \implies A$ and $A \implies B$ (logically equivalent to $\neg B \implies \neg A$), we arrive at $A \iff B$, i.e. $A$ is true if and only if $B$ is true.

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When it comes to "if-then", always draw circles

In math, you have Necessary and Sufficient conditions. Sufficient implies the necessary. This implication is denoted by S => N. When you are inside S, then, for sure you are inside N. When A is sufficient for B, you say "if A then B" and write A => B. When both imply each other, you say "iff A then B" or "iff B then A" because implication is bidirectional and you write equivalence instead of implication, A <=> B.

I think saying that things are equivalent is less confusing than iff.

In

$$8x + 2x - 44 = 220 + 4 - x \implies 11x = 268 \implies x = 24.36$$

It is legal to use $\implies$ despite of equivalence because you show the direction of your derivation. But, I am not mathematician. And I really find your question interesting. I think that professional proofs use turnstile, ⊢, for your implication instead:

$$8x + 2x - 44 = 220 + 4 - x ⊢ 11x = 268 ⊢ x = 24.36$$

to show how first truth implies the other.

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$+1$ for the giant diagram. – Lucas May 26 '13 at 20:20

$\rightarrow$ means implies, not iff; that would be $⟺$.

$A \rightarrow B$ translates in English to "if $A$ is true, then $B$ is true."

$A ⟺ B$ means $A \rightarrow B$ and $B \rightarrow A$ at the same time (and so translates to "if $A$ or $B$ is true, then the other one is true too."

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When is that useful to use? I am really confused by the difference from these both. – user73230 May 26 '13 at 15:39
@user73230, they're both useful. Plus, he already explained the difference. – JMCF125 Jul 21 '13 at 22:06