# Congruence, Equal, and Equivalence

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

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 $$n\in \mathbb{Z}$$.

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.