## First Draft

*I hope to have another draft with drawings to help readability and visualization, especially considering the prominent use of color schemes below. I also would like to bring in some of the interactions I had with students between and during each step.*

**Description**:

An illustration of the formal definition of a relation on a set $A$ as a subset of the cartesian product of $A \times A$, the cartesian product of $A$ with itself; equivalently, a relation on a set $A$ is an element of $\mathcal{P}(A \times A)$, the power set of the cartesian product of $A$ with itself.

To write out in $\LaTeX$ what occurs on the board, we will let $Y$ mean the yellow sticky note and use $B$ for the blue sticky note.

**Ingredients in play**

- sticky notes of two colors
- chalk and chalkboard
- care, creativity, and a lil bit of crazy (<–what others might say)

**Step 0**

What is important about defining a relation on a set and sharing that with anyone, students, math majors?

**Step 1**

Groups! Not yet *those groups*, but that could be part of the thinking process in **Step 0**. Each group of students names themselves and chooses a set name.
The set will comprise two elements, a sticky note of each color:

$$ \mathrm{YB} = \{Y, B \} $$

**Step 2**

We now define the cartesian product of $\mathrm{YB}$ with $\mathrm{YB}$:

Note that the cartesian product (notation: $\times$) of set $S$ with set $T$, whether $S$ and $T$ are the same set or distinct sets, is another set. The cartesian product of $S$ and $T$ is the set of all ordered pairs with a first entry from $S$ and second entry from $T$: $\{(s,t) \mid s \in S,~ t\in T \}$.

In our case, the set $\mathrm{YB}$ is playing the role of both $S$ and $T$ in the above description.

So we have

$$ \mathrm{YB} \times \mathrm{YB} = \{(B,B), (B,Y), (Y,B), (Y,Y) \} $$

**Step 3**

We now define the power set $\mathcal{P}(\mathrm{YB} \times \mathrm{YB})$ of the cartesian product $\mathrm{YB} \times \mathrm{YB}$:

The power set of a set $S$ is another set, truly. It contains all subsets of set $S$. In our case, $\mathrm{YB} \times \mathrm{YB}$ plays the role of $S$:

$$ \mathcal{P}(\mathrm{YB} \times \mathrm{YB}) = \left\{ \begin{array}{l} \emptyset, \{(Y,Y) \}, \{(B,B) \}, \{(B,Y) \}, \{(Y,B) \}, \{(B,B), (Y,Y)\}, \{(B,B), (B,Y)\} \\\
\{(B,B), (Y,B)\}, \{(Y,Y), (B,Y)\}, \{(Y,Y), (Y,B)\}, \{(B,Y), (Y,B)\} \\\
\{(B,B), (B,Y), (Y,B)\}, \{(B,B), (B,Y), (Y,Y)\}, \{(B,B), (Y,B), (Y,Y)\}, \{(B,Y), (Y,B), (Y,Y)\} \\\
\{(B,B), (Y,Y), (B,Y), (Y,B)\}
\end{array}

\right\} $$

**Step 4**

A relation $R$ on the set $\mathrm{YB}$ is an element of $\mathcal{P}(\mathrm{YB} \times \mathrm{YB})$. In other words, a relation $R$ on a set $S$ is a subset of $S \times S$; thus, a relation $R$ on a set $S$ is a set of ordered pairs with both entries from $S$. Note: We can define a relation between two sets $S$ and $T$ by constructing $\mathcal{P}(S \times T)$ and choosing an element (which is a subset of $S \times T$).

**Remarks**
The empty set in **Step 3** corresponds to the empty relation on $\mathrm{YB}$.

Writing $aRb$ means $(a,b) \in R$. “The element $a$ is related to the element $b$.”
Notation can help point to commonly used relations, emphasizing that relations allow us to formally analyze two elements of a set:

- “equal” relation on a set $S$ with notation $=$
- “not equal” relation on a set $S$ with notation $\neq$
- “less than” relation on $\mathbb{R}$ with notation $<$
- “less than or equal to” relation on $\mathbb{R}$ with notation $\leq$
- “greater than” relation on $\mathbb{R}$ with notation $>$
- “greater than or equal to” relation on $\mathbb{R}$ with notation $\geq$

**Definition of a binay relation**
If $S$ and $T$ are two sets, then a *relation* between $S$ and $T$ is a triple $(S,T,R)$ signifying that $S$ is the domain, $T$ is the codomian, and $R$ is a set of ordered pairs with first entries from $S$ and second entries from $T$. This triple shares the data of the sets involved and how their elements are related.

One reason relations are interesting to many people is that *functions* are types of relations. For more details, see my **meaningful symbols**, which contains notes on algebraic constructions and some theoeretic foundations.