preloader

Some (sticky) notes for relations [SNFR]

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:

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

Picture of chalkboard with the following combination of chalk writings and sticky notes: Group name listed as Charlotte; a set named YB defined by placing an actual yellow sticky note comma-separated by an actual blue sticky note within curly brackets.
The Charlotte group named their set $\mathrm{YB}$ after the "yellow-blue" sticky notes.

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

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

Picture of chalkboard with the following combination of chalk writings and sticky notes: As before, there is the Group name listed as Charlotte; a set named YB defined by placing an actual yellow sticky note comma-separated by an actual blue sticky note within curly brackets; and, the cartesian product of YB with itself, which has four ordered pairs between curly brackets. The first ordered pair contains two blue sticky notes; the second ordered pair has a blue sticky note in the first component and a yellow sticky note in the second component; the third ordered pair is the reverse of the second ordered pair; lastly, the fourth ordered pairs has two yellow stickyn notes.
The Charlotte group then formed the cartesian product $\mathrm{YB} \times \mathrm{YB}$ of the set they defined in Step 1.

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 $S$:

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

Picture of chalkboard with the following combination of chalk writings and sticky notes: As before, there is the Group name listed as Charlotte; a set named YB defined by placing an actual yellow sticky note comma-separated by an actual blue sticky note within curly brackets; the cartesian product of YB with itself; and, the power set of the cartesian product of YB with itself, which has all subsets of YB cross YB.
The Charlotte group then formed the power set $\mathcal{P}(\mathrm{YB} \times \mathrm{YB})$ of $\mathrm{YB} \times \mathrm{YB}$. The last subset containing all ordered pairs is not pictured.

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.