MathematicsChapter 3: Sets and Relations
Pairs Products Relations
An ordered pair (a, b) cares about order: (a, b) = (c, d) if and only if a = c and b = d.
Learning objectives
- Explain Pairs Products Relations in simple mathematical language.
- Apply ordered pairs and a x b correctly.
- Apply relation and function correctly.
- Use the symbols, conditions, and formulae accurately.
- Solve a short example and check the answer.
- Recognise and correct common mistakes.
Definition
An ordered pair (a, b) cares about order: (a, b) = (c, d) if and only if a = c and b = d. So (1, 2) is not (2, 1). The Cartesian product A x B is the set of all ordered pairs with first entry from A and second from B. Its size is n(A x B) = n(A)n(B). If A or B is empty, A x B is empty. A relation from A to B is any subset of A x B. A function is a special relation: every domain element is paired with exactly one range value.
Formula
a = c and b
Where / symbols
- ordered pair
- An ordered pair (a, b) cares about order: (a, b) = (c, d) if and only if a = c and b = d.
- (a, b) = (c, d)
- An ordered pair (a, b) cares about order: (a, b) = (c, d) if and only if a = c and b = d.
- a = c
- An ordered pair (a, b) cares about order: (a, b) = (c, d) if and only if a = c and b = d.
- b = d
- An ordered pair (a, b) cares about order: (a, b) = (c, d) if and only if a = c and b = d.
- Cartesian product
- The Cartesian product A x B is the set of all ordered pairs with first entry from A and second from B.
- n(A x B) = n(A)n(B)
- Its size is n(A x B) = n(A)n(B).
- empty
- If A or B is empty, A x B is empty.
- relation
- A relation from A to B is any subset of A x B.
- subset
- A relation from A to B is any subset of A x B.
- A x B
- The Cartesian product A x B is the set of all ordered pairs with first entry from A and second from B.
Relationship
- (a, b) = (c, d)
- a = c
- b = d
- n(A x B) = n(A)n(B)
Overview
Pairs and products first, then relations and a glimpse of functions:
ORDERED PAIRS AND A x B
Equality of pairs needs both coordinates: (a, b) = (c, d) iff a = c and b = d. Cartesian product lists every pairing. If A = {1, 2} and B = {x, y}, then A x B has 2 x 2 = 4 pairs. The formula n(A x B) = n(A)n(B) is just counting independent choices. If A is empty or B is empty, there are no pairs, so the product is empty.
Examples (1, 2) != (2, 1) · n({1, 2} x {a, b, c}) = 6 · {} x {1, 2} = empty
Relation and Function
Any subset of A x B is a relation from A to B. You may give it as a list of pairs or as a rule such as 'a is less than b.' A function is a relation that assigns each element of the domain exactly one output in the range. Not every relation is a function: (1, 2) and (1, 3) together fail uniqueness.
Examples {(1, a), (2, a)} is a relation · Domain: first components used · Range: second components used
WHAT — Core idea
An ordered pair (a, b) cares about order: (a, b) = (c, d) if and only if a = c and b = d. So (1, 2) is not (2, 1). The Cartesian product A x B is the set of all ordered pairs with first entry from A and second from B. Its size is n(A x B) = n(A)n(B). If A or B is empty, A x B is empty. A relation from A to B is any subset of A x B. A function is a special relation: every domain element is paired with exactly one range value.
WHY — Why it matters
A x B is not B x A unless the two pair-sets happen to match. Relations can be empty (the empty subset). For a function, every domain member appears exactly once as a first coordinate; repeats of the first coordinate with different seconds are forbidden.
HOW — Method and examples
1. ORDERED PAIRS AND A x B: Equality of pairs needs both coordinates: (a, b) = (c, d) iff a = c and b = d. 2. RELATION AND FUNCTION: Any subset of A x B is a relation from A to B.
HOW — Answer in an exam
(a, b) = (b, a) only when a = b. n(A x B) = n(A)n(B) fails if you list pairs without order. A relation is a subset of A x B, not of A union B. Functions need a unique output, not merely 'some pairing.'
Explanation
A x B is not B x A unless the two pair-sets happen to match. Relations can be empty (the empty subset). For a function, every domain member appears exactly once as a first coordinate; repeats of the first coordinate with different seconds are forbidden.
In simple words
(a, b) = (b, a) only when a = b. n(A x B) = n(A)n(B) fails if you list pairs without order. A relation is a subset of A x B, not of A union B. Functions need a unique output, not merely 'some pairing.'
Key points
- ordered pair
- (a, b) = (c, d)
- a = c
- b = d
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Pairs Products Relations
Sticky Notes · Quick Revision
WHAT
- An ordered pair (a, b) cares about order: (a, b) = (c, d) if and only if a = c and b = d.
Key Points
- An ordered pair (a, b) cares about order: (a, b) = (c, d) if and only if a = c and b = d.
- Equality of pairs needs both coordinates: (a, b) = (c, d) iff a = c and b = d.
- Any subset of A x B is a relation from A to B.
- A x B is not B x A unless the two pair-sets happen to match.
HOW
- 1. ORDERED PAIRS AND A x B: Equality of pairs needs both coordinates: (a, b) = (c, d) iff a = c and b = d.
- 2. RELATION AND FUNCTION: Any subset of A x B is a relation from A to B.
Examples
- (1, 2) != (2, 1)
- n(1, 2 x a, b, c) = 6
- x 1, 2 = empty
- {(1, a), (2, a)} is a relation
- Domain: first components used
Common Mistake
- (a, b) = (b, a) only when a = b.
- n(A x B) = n(A)n(B) fails if you list pairs without order.
Exam Tip
- (a, b) = (b, a) only when a = b. n(A x B) = n(A)n(B) fails if you list pairs without order. A relation is a subset of A x B, not of A union B. Functions need a unique output, not merely 'some pairing.'
60-second revision
- An ordered pair (a, b) cares about order: (a, b) = (c, d) if and only if a = c and b = d.
- Equality of pairs needs both coordinates: (a, b) = (c, d) iff a = c and b = d.
- Any subset of A x B is a relation from A to B.
- A x B is not B x A unless the two pair-sets happen to match.
- (a, b) = (b, a) only when a = b.
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