Relations

From everyday relations to mathematics. "How is he related to you?" ("my father", "my teacher", "not related") shows that relation connects one person to another. Mathematics borrows this: a relation connects one mathematical object to another -- e.g. "$m$ is related to $n$ if $m$ divides $n$", "$x$ is related to $y$ if $x\le y$", "a point $p$ is related to a line $L$ if $p$ lies on $L$", "student $X$ is related to school $S$ if $X$ studies at $S$".

Illustration -- cryptography. A Caesar cipher shifts every letter three places later in the alphabet: "LET US WIN" becomes "OHW XVZ LQ". Written as ordered pairs, $\{(L,O),(E,H),(T,W),(U,X),(S,V),(W,Z),(I,L),(N,Q)\}$ is a subset of $C\times D$ where $C=\{L,E,T,U,S,W,I,N\}$ and $D=\{O,H,W,X,V,Z,L,Q\}$ -- a relation described purely as a set of ordered pairs.

Illustration -- geometry. The line $2x-y=0$ (i.e. $y=2x$) is the set $\{(x,2x):x\in R\}\subseteq R\times R$; the parabola $x^2-y=0$ is $\{(x,x^2):x\in R\}$; the sideways parabola $x-y^2=0$ splits into $\{(x,\sqrt x)\}$ and $\{(x,-\sqrt x)\}$ for $x\ge0$. All three are subsets of a Cartesian product -- exactly the pattern a relation follows.

Formal definition. For non-empty sets $A,B$, a relation $R$ from $A$ to $B$ is any subset of $A\times B$, written $R\subseteq A\times B$. A relation from $A$ to $B$ is generally different from a relation from $B$ to $A$.

• Domain of $R$: $\{a\in A:(a,b)\in R\text{ for some }b\in B\}$ -- the set of first coordinates that actually occur.
• Range of $R$: $\{b\in B:(a,b)\in R\text{ for some }a\in A\}$ -- the set of second coordinates that actually occur. $B$ itself is called the co-domain; the range is always a subset of the co-domain.

Extreme relations. For any set $A$, both $\varnothing$ and $A\times A$ are relations on $A$: the empty relation and the universal relation. …