Relations and Functions

An ordered pair $(a, b)$ is two objects written in a fixed order — a first component $a$ and a second component $b$. Unlike a set, order is essential: $(a, b) = (c, d)$ only if $a = c$ and $b = d$. So $(2, 3) \ne (3, 2)$, whereas $\{2,3\} = \{3,2\}$.

The Cartesian product $A \times B$ is the set of all ordered pairs whose first component comes from $A$ and second from $B$:

Think of it as every possible pairing — one element of $A$ matched with one element of $B$, exhaustively. Because each of the $n(A)$ first components can be paired with each of the $n(B)$ second components, the size multiplies:

Key properties:

The familiar $xy$-plane is exactly $\mathbb{R} \times \mathbb{R}$: every point is an ordered pair of real numbers. This is why the chapter matters — coordinate geometry, graphs of functions and the whole of calculus live inside a Cartesian product.

Deeper Insight — why "ordered" changes everything: The single word ordered is what separates this chapter from the previous one on sets. A set answers only "is this object in or out?"; an ordered pair additionally records "in which role?" — first or second, input or output, $x$ or $y$. That extra bit of structure is precisely what lets a Cartesian product model relationships between two collections rather than just a lump of objects. Once you grasp that $A \times B$ lists every conceivable pairing, the next two ideas fall out naturally: a relation is just a chosen subset of those pairings (the ones that actually satisfy some rule), and a function is a relation disciplined by one extra condition. Everything downstream — domain, range, graphs, even the coordinate plane you have used since Class 9 — is built on this one move from unordered membership to ordered pairing.

A relation $R$ from set $A$ to set $B$ is any subset of $A \times B$. So a relation simply selects the ordered pairs that satisfy some rule, leaving the rest out:

For each relation we name three sets:

The range is always a subset of the codomain ($\text{range} \subseteq B$), but they need not be equal — the codomain is the "menu", the range is "what was ordered".

Counting relations: a relation is a subset of $A \times B$, which has $n(A)\cdot n(B)$ pairs. Since a set with $k$ elements has $2^k$ subsets, the number of possible relations from $A$ to $B$ is

Relations are often given by a rule, e.g. $R = \{(x, y) : y = x + 1,\ x \in A\}$, which you then list in roster form by testing each input.

Deeper Insight — a relation is a filter, and that is the bridge to functions: The whole content of this idea is "subset of the Cartesian product". $A \times B$ offers every possible pairing; a relation keeps only the pairings that obey a chosen condition — it is a filter on pairings. Reading the definition this way makes the domain/range distinction obvious: the domain is which inputs survived the filter, the range is which outputs they map to. It also explains the codomain, the subtlest of the three: we declare a target set $B$ in advance (the codomain) before we know which of its members will actually be hit (the range). This advance declaration is exactly what makes the next section possible — a function is nothing more than a relation with one extra rule imposed on the filter ("each input is used once, and exactly once"), so understanding a relation as a constrained subset is the conceptual key to the rest of the chapter.

A function $f$ from $A$ to $B$ is a relation in which every element of $A$ is paired with exactly one element of $B$. Two demands, both essential: every input is used (no element of $A$ left out), and each input gives one output (never two). We write $f : A \to B$ and $y = f(x)$.

The vertical line test captures this graphically: a curve is a function exactly when no vertical line meets it more than once (one $x$, one $y$).

Standard real functions you must recognise by formula and graph:

Algebra of real functions: for functions with overlapping domains you may add, subtract, multiply and divide them pointwise: $(f \pm g)(x) = f(x) \pm g(x)$, $(fg)(x) = f(x)g(x)$, and $\left(\tfrac{f}{g}\right)(x) = \tfrac{f(x)}{g(x)}$ wherever $g(x) \ne 0$.

Finding the domain of a real function means listing every $x$ for which the rule gives a real value — exclude inputs that cause division by zero or an even root of a negative number. The range is the resulting set of outputs.

Deeper Insight — "exactly one output" is the entire idea, and it is what makes calculus possible: A function is a relation that has been disciplined by a single rule: one input, one output. That rule looks modest but it is the reason functions, not arbitrary relations, became the central object of mathematics — because a unique output per input is exactly what lets us speak of "the value of $f$ at $x$", differentiate it, integrate it, and invert it. The vertical line test is just this rule drawn on a graph. The standard functions in the table are worth memorising not as trivia but as a vocabulary: nearly every function you meet later is built by combining, transforming or composing these few, so knowing their shapes (the V of $|x|$, the steps of $[x]$, the three-level jump of the signum) lets you predict the behaviour of complicated expressions at a glance. Master "unique image" and the standard graphs, and the leap to Limits and Derivatives in Chapter 12 becomes a short step rather than a cliff.