Sets, Relations and Funcs

October 4, 2020

This page discusses Sets,Relations and Function. The ideas presented here only skim the surface. Definitions and discussion follow.

relation

A subset $R$ of $A \times B$ is a relation from $A$ to $B$.

inverse

If $R$ is $\{(a, b) | (a,b) \in R\}$, then $R^{-1} = \{(b, a) | (a, b) \in R\}$

function

a mapping of $f$ from $X$ into $Y$ such that if $(x,y_1) \in f$ and $(x,y_2) \in f$, then $y_1=y_2$. To designate this we write $f:X \rightarrow Y$.

onto

if f is a function from $X$ to $Y,$ and the range(see below) of $f$ is $Y$ then $f$ is onto.

one-to-one

a function is one-to-one if $(x_1,y) \in f$ and $(x_2,y) \in f$ implies $x_1=x_2.$ </dl>

The '$\times$' symbol means cartesian product. Also, if $A_1 \subset A, R(A_1) = \{b \in B | (a, b) \in R$ for some $a \in A_1\}. R(A_1)$ is called the image of $A_1$ under $R$. Likewise, if $B_1 \subset B, R^{-1}(B_1) = \{a \in A | (a,b) \in R$ for some $b \in B_1\}. R^{-1}$(B_1) is called the inverse image of $B_1$ under $R$. The inverse image of $B$ is called the domain of $R$ and is denoted $D_R$. The image of A under $R$ is called the range and is denoted $R_R$. If $X$ is a relation from $A$ to $B,$ and $Y$ is a relation from $B$ to $C,$ $X \circ Y = \{(a, c) |$ for some $b, (a, b) \in X$ and $(b, c) \in Y\}$. The empty set is a relation for any set $A$ to $B$.

The less than symbol, $<$ , creates a relation on the coordinate axes. All the points above the line $y = x$ are elements of this relation. Likewise the relation, $>$ , creates another relation on the coordinate axes. All the points below the line $y = x$ are members of this relation. We can also have relations with subsets of the coordinate axes.

Suppose $A = (2,4)$ and $B = (2,4).$ The relation $A \times B$, are the set of points in the first quadrant of the coordinate plane inside of the square formed by the points $(2,2), (4,2), (4,4)$ and $(2,4).$