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:
$$8x + 2x - 44 = 220 + 4 - x \implies 11x = 268 \implies x = 24.36$$
$\endgroup$ 97 Answers
$\begingroup$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 $-1\not> 0$)
$r=1\iff r+1=2$
The Wikipedia page on logical connectives should be helpful.
$\endgroup$ 3 $\begingroup$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 $B$".
$\endgroup$ $\begingroup$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.
$\endgroup$ 2 $\begingroup$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
$\endgroup$ $\begingroup$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.
$\endgroup$ $\begingroup$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.
$\endgroup$ 1 $\begingroup$$\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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