Pairs Products Relations

03Adjust the depth

Choose Your Level

What is it?

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.

In simple words

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.

Important terms

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.

What you should learn

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.

Examples

(1, 2) != (2, 1)
n(1, 2 x a, b, c) = 6
x 1, 2 = empty

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.

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.

Full school-level explanation

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.

Pairs and products first, then relations and a glimpse of functions:

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.'

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.

Remember

(a, b) = (c, d)
a = c
b = d
n(A x B) = n(A)n(B)
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.

Remember

subset
A x B
relation
function
domain
range

Examples

{(1, a), (2, a)} is a relation
Domain: first components used
Range: second components used

Remember

ordered pair
(a, b) = (c, d)
a = c
b = d
Cartesian product
n(A x B) = n(A)n(B)
empty
relation

How the ideas connect

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.

04Exam readiness

Exam Focus

Important points

ordered pair
(a, b) = (c, d)
a = c
b = d

Important formulas

ordered pair
(a, b) = (c, d)
a = c

Common exam traps

(a, b) = (b, a) only when a = b.
n(A x B) = n(A)n(B) fails if you list pairs without order.

Long question structure

Definition → key parts → example → exam caution

05Watch out

Common Mistakes

Mistake
(a, b) = (b, a) only when a = b.
Mistake
n(A x B) = n(A)n(B) fails if you list pairs without order.
Mistake
A relation is a subset of A x B, not of A union B.
Mistake
Functions need a unique output, not merely 'some pairing.
Mistake
(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