How is = defined?

Question

What does it mean to be equal? $x$ is equal $y$, written symbolically $x=y$. I know that $=$ is a equivalence relation. I have never seen a formal definition of the relation $=$.

Answer 1

The first time you formally encounter $=$ will probably be in set theory, where it is a relation on sets defined as $A=B$ iff all members of $A$ are equal to all members of $B$. (And of course, the empty set equals itself).

More generally, the equality operator needs to be defined on members of sets in order to match the goals of the work you intend on doing. An equality operator on set $X$ is a binary operator, i.e., it is a function $f:X\times X\to \{=,\neq\}$, where any member $(x_1,x_2)\in f^{-1}(=)$ has the property that $x_1=x_2$. Mathematicians have introduced three criteria for a consistent notion of equality:

1. Reflexive: $x=x$ for all $x\in X$.
2. Symmetric: If $x=y$, then $y=x$.
3. Transitive: If $x=y$ and $y=z$, then $x=z$.