An ordered pair $(a, b)$ is a pair of objects in which the order matters: $a$ is the first component and $b$ is the second component. This is the crucial difference from a set — $\{a, b\} = \{b, a\}$, but $(a, b) \ne (b, a)$ unless $a = b$. An ordered pair records not just which two objects appear, but which comes first.
Equality of ordered pairs. Two ordered pairs are equal only when both components match in their respective positions:
This single rule solves every "find $x$ and $y$" problem in this exercise: equate first component to first, second to second, and solve the two resulting equations.
The Cartesian product. Given two non-empty sets $A$ and $B$, the Cartesian product $A \times B$ (read "$A$ cross $B$") is the set of all ordered pairs whose first component comes from $A$ and second from $B$:
You build it by pairing every element of $A$ with every element of $B$ in turn — like filling in a grid with rows labelled by $A$ and columns by $B$. The name honours René Descartes, whose coordinate plane $\mathbb{R} \times \mathbb{R} = \mathbb{R}^2$ is exactly this construction with $A = B = \mathbb{R}$.
The counting rule. Because each of the $n(A)$ first components can be paired with each of the $n(B)$ second components, the total number of ordered pairs is the product:
This is the fundamental counting principle in disguise, and it extends to any number of sets. In particular $n(A \times A) = \big(n(A)\big)^2$ and $n(A \times A \times A) = \big(n(A)\big)^3$.
Products of a set with itself. Two short-hands appear constantly. $A \times A$ (also written $A^2$) is the set of all ordered pairs from $A$; and $A \times A \times A$ (or $A^3$) is the set of all ordered triples $(a, b, c)$ with each component in $A$. An ordered triple obeys the same position-matters rule: $(a, b, c) = (d, e, f)$ only when $a=d$, $b=e$ and $c=f$.
Product with the empty set. By convention, if either set is empty there are no pairs to form, so:
Non-commutativity. Unlike union and intersection, the Cartesian product is not commutative: in general $A \times B \ne B \times A$. The pair $(1, a)$ from $A \times B$ is a different object from $(a, 1)$ in $B \times A$. The two products are equal only in the special cases $A = B$, or when one of the sets is empty. Note the counts always agree — $n(A \times B) = n(B \times A)$ — but equal cardinality does not mean equal sets.
| Property | Statement |
|---|---|
| Definition | $A \times B = \{(a,b) : a \in A,\ b \in B\}$ |
| Count | $n(A \times B) = n(A)\,n(B)$ |
| Self-product | $n(A \times A) = n(A)^2$, $n(A^3) = n(A)^3$ |
| Empty set | $A \times \varnothing = \varnothing$ |
| Commutativity | $A \times B \ne B \times A$ in general |
Deeper Insight — the ordered pair is what makes coordinates, relations and functions possible: Everything in this whole chapter — and a great deal of the rest of mathematics — is built on the humble ordered pair. The reason "order matters" is not a technicality: it is precisely what lets $(3, 5)$ name a single point on the plane distinct from $(5, 3)$, what lets a relation record "$x$ is related to $y$" as a directed link rather than a symmetric one, and what lets a function send each input to exactly one output. The Cartesian product is simply the universe of all possible pairings; a relation, which you meet next, is just a chosen sub-collection of that universe. Once you see $A \times B$ as the full grid of possibilities, a relation becomes "the cells you tick" and a function becomes "exactly one ticked cell per row" — and the entire chapter clicks into place.