We might arguably say that mathematics is the study of how various entities are related; in any case, the relationships between mathematical objects is a large part of what we study. You are already familiar with many such relationships: If $f(x)=y$ then $x$ and $y$ are related in a special way; if we say $x< y$ or $x=y$ or $x\ge y$, we are highlighting a particular relationship between $x$ and $y$.

Certain kinds of relationships appear over and over in mathematics,
and deserve careful treatment and study. We use the notation $x\sim y$
to mean that $x$ and $y$ are related in some special way; "$\sim$''
is called a **relation**. The meaning of $\sim$
changes with context—it is not a fixed relation. In some cases, of
course, we can use other symbols that have come to be associated with
particular relations, like "$< $'' and "$=$''. We could give a
formal definition of the term **relation**, but for our
purposes an intuitive approach will be sufficient, just as we have
made do without a formal definition of "function''.