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If someone asks you to solve 2x+4=102x + 4 = 10, you get exactly one answer: x=3x = 3. But what if you are asked to solve 2(x+2)=2x+42(x + 2) = 2x + 4?
Every real number you plug in makes it true. In mathematics, this difference between finding specific unknown values and stating universal structural truths is the heart of distinguishing equations, identities, and functions.

What is an Identity?

An identity is an equality between two mathematical expressions that holds true for all possible values of the variables in their shared domain.
To show that two expressions are identical rather than just equal at specific points, we use the triple-bar symbol ≑\equiv, which reads "is identically equal to."
Identities describe the equivalence of algebraic structures, but what tool describes a rule that transforms inputs into outputs?

What is a Function?

A function is a relation that pairs each element in an input set, called the domain, with exactly one element in an output set, called the codomain.
We write this mapping formally as f:XoYf: X o Y, where the formula f(x)=yf(x) = y defines the operation applied to any input xx to yield its unique image yy.
Now that we understand mappings and universal equalities, how do we distinguish them when analyzing expressions in formal proofs?

Classifying Mathematical Statements