Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Some texts and online sources seem to imply that they're interchangeable, while others say that there is at least a small nuanced difference between the two.

For example, I've read that $\equiv$ can be read as "can be replaced in a logical proof with...", yet I've also seen $\leftrightarrow$ used in that same vain.

Edit: I'm trying to write my own textbook on this material as a way of forcing me to learn as much as possible before my final in 3 weeks. I'm trying to figure out a way to describe what $\equiv$ means, while I've described $\leftrightarrow$ as "If and only if"

share|cite|improve this question
It means whatever your lecturer says it means. – Michael Greinecker Nov 18 '12 at 0:48
As far as i've seen, they are equivalent and iff respectively. Mean the same but emphasis is different. – Inquest Nov 18 '12 at 0:51
If $=$ means "equal", then $\equiv$ means "really equal, since it has one more horizontal bar. :-) – coffeemath Nov 18 '12 at 1:51
@coffeemath like $<<$ means really smaller than and $>>$ means really larger than? ;-) – amWhy Nov 18 '12 at 2:13
@amWhy: Between all things that are equal, some things equal more than others! :-) – Asaf Karagila Nov 18 '12 at 2:22
up vote 13 down vote accepted

In propositional calculus there is a connective $\rightarrow$ and often one for $\leftrightarrow$, which is a part of the formal language and how we create new propositions; and there is another form, $\implies$ and $\iff$ (also written as $\cong$) which are meta-statements.

Whereas $\varphi\leftrightarrow\psi$ is a proposition, $\varphi\iff\psi$ is a statement about propositions.

It is true, however, that $\varphi\iff\psi$ is true, if and only if $\varphi\leftrightarrow\psi$ is a tautology. However there is a difference between a proposition, and a statement about propositions; and I cannot stress this difference enough in this post.

share|cite|improve this answer
Snow is white if and only if it is true that snow is white? – Neal Nov 18 '12 at 1:52
@Neal: Yes. Exactly. One is a predicate (being white) about a term (snow), the other is a claim about the term and the predicate. – Asaf Karagila Nov 18 '12 at 1:57
@Neal The point is that in general some meta-statements cannot be expressed (or proved) inside a logic system, therefore we have to be very careful if we're "inside" or "outside" (at the meta- level). – Petr Pudlák Nov 20 '12 at 1:16

With respect to the "$\equiv$" symbol (my comment was getting long!):

This is more in the way of speculation, but it seems that "$a \equiv b$" is used in some (not all) contexts to denote "$a$ is identically $b$", or in other contexts to convey that $a$ and $b$ are essentially equivalent (with respect to some equivalence-relation, e.g. congruence modulo $n$, or geometric congruence, or truth-functionality, or... etc.), again, depending on the contexts in which it's being used.

This would be consistent with the frequent use of the symbol "$\equiv$" in introductory logic texts to convey that $a \equiv b$ holds (is true) whenever $a$ and $b$ evaluate to the same truth value. It's also consistent with what you've read: "$\equiv$" can be read as "can be replaced in a logical proof with...", in the sense that if "$p\equiv q$", then replacing every occurrence of $p$ with $q$ (or vice-versa) will not change the truth-value of any propositions thus impacted.

I know that I've used "$\equiv$" and "$\leftrightarrow$" interchangeably on this site, when answering, e.g., questions about propositional logic: partly for reasons related to trying to match the notation used in the question, and partly due to being careless and/or not necessarily knowing better.

share|cite|improve this answer

Some author as do use $\equiv$ to mean "if and only if". Another option, which I use from time to time, is to use $\equiv$ as a sort of equality symbol when I am talking about formulas. If I want to give the name $\phi$ to the formula "$a=b"$, I don't want to write $$ \phi = a = b $$ so I write $$ \phi \equiv a = b $$ instead. This allows me to avoid using quotation symbols.

This latter usage is not documented anywhere, I as far as I know. I am not aware of any book that describes it in detail. But I can explain my motivations for using it:

  • I don't want to re-use the $=$ sign for another purpose; the first displayed formula is not appealing. But quotation marks in displayed formulas are also unappealing.

  • When I write $\phi \equiv a = b$, I sometimes don't care if $\phi$ is the formula $a=b$ or is just equivalent to it. But for the rest of the proof I will act as if it is that formula. So the $\equiv$ symbol, which connotes "equivalent", suggests that all that matters in the beginning is that $\phi$ is equivalent to $a=b$.

share|cite|improve this answer

"↔" denotes an equivalent statement in formal logic.

e.g. A^B ↔ B^A (because of the commutative properties).

While "≡" denotes an equivalent statement in a mathematical equation, but definition wise it means "identical to".

e.g. 3*4 ≡ 4*3 (because of the commutative properties).

While like others said these can be interchangeable because they do mean the same thing you usually see "↔" in formal logic and "≡" in mathematical equations.

I should also note this perspective is from a CS major and may vary from a mathematical majors perspective, but this is what I was taught.

You may want to check this out also as it is the book I am currently using for my Discrete Structures class:

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.