Relations

We introduced the idea of Cartesian products on slide 18 of Topic 2 and said that they are sets of all the possible ordered pairs of the form $(s, t)$, where $s \in S$ and $t \in T$ ($S$ and $T$ are sets of some kinds of elements.) The notation for a cartesian product is $S \times T$ (

$S \times T$

).

In our chapter, we will present various sets of ordered pairs. In particular, a relation (also called a binary relation,) denoted by $R$ or some other symbol or name, is a set of ordered pairs of the form $(s, t)$, where $s \in S$ and $t \in T$. It doesn't necessarily contain all the possible pairs $(s, t)$. That is, $R$ is a subset of $S \times T$, and not necessarily equal to $S \times T$. Examples:

• For $S = \{$😀$,$ 😁$,$ 😑$,$ 😒$\}$ and $T = \{😔, 😄, 😆\}$, we can create the relation:
  $\text{Smiles} = \;$$\{(😀, 😄), \,$$(😀, 😆), \,$$(😁, 😄), \,$$(😁, 😆)\}$.
• For $\text{Grades} = \{85, 90, 95, 100\}$ and $\text{Letters} = \{B-, B, B+, A-, A, A+\}$, we can create the relation:
  $\text{CUNYfirst-Grades} = \;$$\{(85, B), \,$$(90, A-), \,$$(95, A), \,$$(100, A+)\}$.