Relating Things Together
Through the use of functions, we can define a lot of the things that we use in general, such as operations like $+$ and $\cdot$. As of right now though, we don’t really have a good way of comparing things together. For example, how would we define a simple comparison like $x < y$? This is an important question as literally all the time we want to be relating and comparing things to each other; if you’re a Java programmer you’ll know about these as comparators.
One way that we can do it is to define a function $< : \mathbb{R}\rightarrow\{0,1\}$ and say that an output of $1$ represents TRUE and $0$ represents FALSE. This is totally fine, but a bit clunky and would require some of our properties we will show in the future would be hella fucking annoying to prove in this way. As such we will create a special type of set to do the work for us!
Definition: A relation $R \subseteq A_1\times A_2\times\ldots\times A_n$ is a set of tuples $t$ for which we say that if $t\in R$ then the items in $t$ are related, and if $t\not\in R$ then the items in $t$ are not related. If $*$ is an operation that relates two items together then $R(*)\subseteq A\times B$ is a binary relation and $a*b$ is TRUE if and only if $(a,b)\in R(*)$.
In this sense, to define a way to related items together, you throw all the groups of items that are properly related in a set, and in order to check if the relation $a*b$ is true (where $*$ represents a general symbol) then just check if $(a,b)$ is a set member. For our cases we will be considering only binary relations, but this can be much more general.
Examples of Relations
In this section we will provide some examples of relations that we will use on the next page in the discussion of properties of relations.
Example: Let $R(=)=\{(a,b)\in A\times A | \text{a,b are the same}\}$. In this case $=$ is standard equality.
For standard equality we would have that $1=1$ but $1\neq 2$. Pretty much just the normal old equality we are used to.
Example: Let $R(\mod n)$ be the set of all pairs of numbers in $\mathbb{Z}$ that have the same remainder when divided by $n$1. This is notated as $a\equiv b\mod n$.
Let $n=2$. Since both $6$ and $4$ are divisible by $2$, then the remainder is $0$ and we say that $$ 6\equiv 4\mod 2. $$ However $3$ is not divisible by $2$, so $$ 6\not\equiv 3 \mod 2. $$
Example: Let $R(\leq)=\{(x,y)\in\mathbb{R}\times\mathbb{R} | x<y\text{ or } x=y\}$. This is just normal $\leq$
As is standard, $1\leq 2$ but $1\not\leq 0$.
Example: Let $p,q$ be people. We will say that $(p,q)\in R(\text{isParent})$ if $p$ is a parent of $q$. We will notate this as $p\text{.isParent}(q)$.
Here we are defining a function that relates two items in a way that we could imagine inside of our code! I have no kids2 so I would not be Joe.isParent() of anyone, but my mom is my parent so Susan.isParent(Joe) would be true.