Congruence, Equal, and Equivalence

Question

I know this is very basic problem about math. But sometimes confusing. What is the difference among

Equal Sign $\left(\,=\,\right)$

Congruence Sign (we saw this on number theory) $\left(\,\equiv\,\right)$

Equivalence Sign $\left(\,\iff\,\right)$

Answer 1

Equals can be generalized to an equivalence relation. This means a relation on a set $S$, $\sim$ which satisfies the following properties:

1. $a\sim a$ for all $a\in S$ (Reflexive)
2. If $a\sim b$, then $b \sim a$ (Symmetric)
3. If $a \sim b$ and $b\sim c$, then $a \sim c$ (transitive).

Equals should satisfy those 3 properties.

Congruence goes one step further. It is used to indicate that it preserves some kind of operation on the set. In your case, congruence mod $n$ is indicating that $a \pmod n$ times $b \pmod n$ is the same thing as $ab \pmod n$. So you can exchange what it is equivalent to before doing the operation or after and you get the same thing. It is also congruence under addition.

$\Leftrightarrow$ is usually talking about the equivalence of two statements. For instance $a \in \mathbb{Z}$ is even if and only if ($\Leftrightarrow$) $a=2n$ for some $n\in \mathbb{Z}$.

Answer 2

The equal sign between two items mean they are the same. Depending the context this equality is defined or assumed to be understood.

For example if $A$ and $B$ are sets, then $A=B$ means every element of $A$ is an element of $B$ and every element of $B$ is an element of $A$.

On the other hand if $a/b$ and $c/d$ are fractions, then $a/b=c/d$ is defined as $ad=bc$

Congruence sign,$\left(\,\equiv\,\right)$ comes with a (mod). The definition $a\equiv b, \pmod {n} $ is that $b-a$ is divisible by $n$

For example $27\equiv 13 \pmod 7$

The $\iff$ sign is if and only if sign and $p\iff q$ means $p$ implies $q$ and $q$ implies $p$ where $p$ and $q$ are statements.