Relations and Functions — Meaning and Definition

Gujarat Std 11 Commerce students meet the idea of a function through the more general idea of a relation between two sets. Understanding this distinction carefully is the foundation for everything else in this chapter — and for the correlation, regression and probability chapters that follow in Std 12 Statistics.

Ordered pairs and the Cartesian product

If $A$ and $B$ are two non-empty sets, the Cartesian product $A \times B$ is the set of all ordered pairs $(a, b)$ such that $a \in A$ and $b \in B$:

For example, if $A = \{1, 2\}$ and $B = \{3, 4, 5\}$, then

Relation

A relation $R$ from set $A$ to set $B$ is simply any subset of $A \times B$. It pairs some (not necessarily all) elements of $A$ with some elements of $B$, following any rule chosen — or no rule at all.

• The domain of a relation is the set of first elements (from $A$) that actually appear in its ordered pairs.
• The range of a relation is the set of second elements (from $B$) that actually appear.

Function

A function is a special kind of relation with two extra conditions:

1. Every element of $A$ must have an image in $B$ (the domain is the whole of $A$, nothing is left out).
2. Each element of $A$ has exactly one image in $B$ (no element of $A$ is paired with two or more elements of $B$).

If $f$ is a function from $A$ to $B$, this is written $f : A \to B$, and if $f$ pairs $x \in A$ with $y \in B$, this is written $y = f(x)$ — read "$y$ equals $f$ of $x$" — with $y$ called the image of $x$ under $f$.

Every function is a relation, but not every relation is a function. A relation fails to be a function if even one element of $A$ is left unpaired, or is paired with more than one element of $B$.

The Gujarat Std-11 Statistics syllabus (Business Mathematics & Statistics) draws on exactly these principles of relations and functions that underlie business mathematics, economics and everyday decision-making — a firm's cost, revenue and demand are all, at their core, functions of quantity or price, as later sections of this chapter show directly.