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Lesson 1 of 5

Relations and Functions · Lesson 1 of 5

Ordered Pairs and Cartesian Products

“Build ordered pairs systematically and see how products of sets describe combinations and coordinates.”

Learning Objectives

• Explain why the order of the entries in an ordered pair matters. • Construct Cartesian products and count their elements. • Use equality of ordered pairs to find unknown entries. • Connect ordered pairs and triplets to coordinates in two and three dimensions. • Reason about Cartesian products involving empty sets, unions, intersections and subsets.

From choices to ordered pairs

Suppose you choose a colour and then an object. A red bag and a blue bag are different combinations, even though both contain a bag. Recording the two choices together gives a precise way to describe the combination. The position of each entry tells us what kind of choice it records.

Let A = {red, blue} and B = {bag, coat, shirt}. Write the colour first and the object second. Starting with red gives (red, bag), (red, coat) and (red, shirt); starting with blue gives three more pairs. Each colour must be paired with every object. This systematic approach prevents both omissions and repeated pairs.

Definition
Ordered pair

An ordered pair (a, b) records a first entry a and a second entry b. Two pairs are equal precisely when their first entries are equal and their second entries are equal.

A set such as {2, 5} does not record order: {2, 5} and {5, 2} are the same set. In contrast, (2, 5) and (5, 2) are different ordered pairs. An ordered pair can also repeat an entry, as in (2, 2). Braces for a set and parentheses for a pair therefore do different jobs.

Equality of ordered pairsLaTeX
The symbol ⇔ means “if and only if”: the statement works in both directions.
Example — Matching corresponding entries

Problem
If (x + 1, y − 2) = (3, 1), find x and y.

  1. 1.Equal first entries give x + 1 = 3. Subtract 1 from both sides to obtain x = 2.
  2. 2.Equal second entries give y − 2 = 1. Add 2 to both sides to obtain y = 3.
  3. 3.Check: (2 + 1, 3 − 2) = (3, 1). We matched positions; we did not compare x + 1 with the second entry 1.

The Cartesian product

Instead of selecting just a few combinations, we can collect all possible ordered pairs from two sets. The first set supplies the first entry and the second set supplies the second entry. This complete collection is called their Cartesian product.

Definition
Cartesian product

The Cartesian product A × B is the set of all ordered pairs (a, b) with a in A and b in B. The symbol × here describes a product of sets, rather than multiplication of their individual entries.

Cartesian productLaTeX
The colon means “such that”; ∈ means “belongs to”. Both membership conditions must hold.

For the colour and object sets, A × B contains six pairs. A pair such as (red, shoes) is excluded because shoes is not in B. A pair such as (bag, red) is excluded because bag is not in A and red is not in B. The definition controls both membership and order.

First entryPairs formed with all second entries
red(red, bag), (red, coat), (red, shirt)
blue(blue, bag), (blue, coat), (blue, shirt)

A similar arrangement explains a simplified vehicle-code example. Take A = {DL, MP, KA} and B = {01, 02, 03}. A code uses a region label first and a number label second. Pairing each region with all three numbers gives nine possible codes. These are abstract labels for combinations; their order is part of the recording rule.

A = {DL, MP, KA}BDLMPKA030201
A Cartesian product as a grid— Each dot is one possible pair. Read the first entry horizontally and the second entry vertically.
Example — Reversing the two sets

Problem
Let P = {a, b, c} and Q = {r}. Find P × Q and Q × P.

  1. 1.For P × Q, take a, b and c in turn as first entries. The only possible second entry is r. Thus P × Q = {(a, r), (b, r), (c, r)}.
  2. 2.For Q × P, r is the first entry and a, b or c is the second. Thus Q × P = {(r, a), (r, b), (r, c)}.
  3. 3.Both products have three elements, but they are different sets of pairs. Equal size does not mean equal sets.
Reversing a product

In general A × B differs from B × A. “In general” does not mean “always”: if A = B, the products are equal. For nonempty sets, equality of the two products actually forces A = B, because each first entry must belong to both sets.

Counting pairs and handling empty or infinite sets

Listing works well for small sets, but counting is more efficient when there are many choices. Fix one first entry and count the possible second entries. Repeat that count for every first entry; the groups of pairs cannot overlap because their first entries differ.

Number of pairsLaTeX
For finite sets, n(A) = p and n(B) = q are their numbers of elements. There are p groups of q pairs.

If A has two elements and B has four, their product has eight ordered pairs. A product with an empty set has no pairs at all: there is no entry available from one required position. This remains true even when the other set is large or infinite.

Empty productsLaTeX
The symbol ∅ denotes the empty set.

If both sets are nonempty and one is infinite, the product is infinite. For example, in {0} × ℝ, every real number t supplies a different pair (0, t). The nonempty condition matters: ∅ × ℝ is still empty. Here ℝ denotes all real numbers, including integers, fractions and irrational numbers.

Example — Recovering the original sets

Problem
A × B = {(p, q), (p, r), (m, q), (m, r)}. Find A and B.

  1. 1.The first entries that occur are p and m, so A = {p, m}.
  2. 2.The second entries that occur are q and r, so B = {q, r}.
  3. 3.Their product has 2 × 2 = 4 pairs, exactly the four given. This recovery works because the product is nonempty; an empty product alone does not determine both original sets.

Products, intersections and unions

A product can combine naturally with the set operations you already know. The important question is still which entries are allowed. An intersection requires a second entry shared by two sets; a union allows a second entry from either set.

Example — Following the allowed second entries

Problem
Let A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}. Compare products with B ∩ C and B ∪ C.

  1. 1.B ∩ C = {4}. Therefore A × (B ∩ C) = {(1, 4), (2, 4), (3, 4)}.
  2. 2.A × B contains (1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4). A × C contains (1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6). Their shared pairs are exactly the three pairs with second entry 4.
  3. 3.B ∪ C = {3, 4, 5, 6}. Pairing these with each entry of A gives twelve pairs. Taking the union of A × B and A × C gives that same collection, counting shared pairs only once.
Products distribute over these operationsLaTeX

These equalities follow from membership, not from a coincidence in the example. A pair (a, b) belongs to the first intersection exactly when a belongs to A and b belongs to both B and C. The union version replaces “both” by “at least one”. Similarly, if A ⊆ B and C ⊆ D, then every pair from A × C is also in B × D. The symbol ⊆ allows equality as well as a smaller subset.

Ordered triplets and coordinate spaces

Two entries locate a point in a plane: one gives the horizontal coordinate and the other the vertical coordinate. Adding a third entry records a third independent coordinate. Products of sets therefore provide a language for locations as well as combinations.

Definition
Ordered triplet

An ordered triplet (a, b, c) has a first, second and third entry. Equality requires equality in all three corresponding positions.

A product of three copies of a setLaTeX
Example — Three independent choices

Problem
For P = {1, 2}, list P × P × P.

  1. 1.Begin with first entry 1: (1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2).
  2. 2.Repeat with first entry 2: (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2).
  3. 3.There are 2 × 2 × 2 = 8 triplets. Entries may repeat because each position is an independent choice from P.

ℝ × ℝ is the collection of all coordinate pairs in a two-dimensional plane. ℝ × ℝ × ℝ is the collection of all coordinate triplets in three-dimensional space. For finite coordinate sets {a₁, a₂} and {b₁, b₂, b₃, b₄}, their product gives a grid of eight points. Swapping coordinates generally moves a point; the entries are positions, not an unordered list.

Quiz

Quick check

Which equality of ordered pairs is correct?

Quick check

A has 4 elements and B has 3 elements. How many pairs are in A × B?

Quick check

If (x − 2, y + 3) = (4, 1), which values are correct?

Quick check

What is ∅ × ℝ?

Quick check

If A = {1} and B = {2, 3}, which set equals A × B?

Quick check

Which statement is always true for finite A and B?

Practice Problems

Practice Problems
  1. Find x and y if (x/3 + 1, y − 2/3) = (5/3, 1/3), and check both entries.
  2. For G = {7, 8} and H = {5, 4, 2}, list both products and explain why they differ.
  3. Let A = {−1, 1}. List all ordered triplets from A × A × A and explain your counting method.
  4. Take A = {1, 2}, B = {2, 3}, C = {3, 4}. Verify the product identities for intersection and union by listing pairs.
  5. A × A has nine elements and contains (−1, 0) and (0, 1). Determine A and list its product. Explain why no fourth element is possible.
  6. Let A = {1, 2}, B = {1, 2, 3}, C = {5} and D = {5, 6}. Explain why A × C ⊆ B × D.
  7. For a four-element product A × B, explain why it has sixteen subsets. Describe how including or excluding each pair constructs a subset.

Key Takeaways

Key Takeaways

• Ordered pairs are equal only when corresponding entries agree. • A × B contains every allowable first–second combination, with n(A)n(B) pairs for finite sets. • A product with an empty set is empty; a nonempty product with an infinite factor is infinite. • Products distribute over union and intersection, and respect subset inclusion. • Ordered pairs and triplets describe coordinates in two and three dimensions.

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Relations, Domain, Range and Codomain