Xjyuk

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 $left(,iff,right)$

There is a subtle difference between $=$ and $equiv$. $equiv$ means that the two sides are ALWAYS equal.

For example:

$$2+2 equiv 4$$

The $=$ is slightly weaker a claim. For example,

$$2x = 4$$

holds only when $x=2$.

Also note that $equiv implies =$ but $= notimplies equiv$

The last $iff$ when two claims imply each other.

For example:

$$2x = 4 iff x=2$$

This essentially means that $2x=4 implies x = 2$ AND $x =2 implies 2x = 4$.

Hope this helps.

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 $aequiv b, pmod n $ is that $b-a$ is divisible by $n$

For example $27equiv 13 pmod 7$

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

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

1. 
   $asim a$ for all $ain S$ (Reflexive)


2. If $asim b$, then $b sim a$ (Symmetric)


3. If $a sim b$ and $bsim c$, 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 $a (mod n)$ times $b (mod n)$ is the same thing as $ab (mod n)$. 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 $a in mathbbZ$ is even if and only if ($Leftrightarrow$) $a=2n$ for some $nin mathbbZ$.