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''.

1. Equivalence Relations

2. Factoring Functions

3. Ordered Sets

4. New Orders from Old

5. Partial Orders and Power Sets

6. Countable total orders