Notes · Unit 3
Sets and Relations
Thirteen short slides on sets, set operations, counting with Venn diagrams, ordered pairs, Cartesian products and relations.
Press Next (or the right arrow key) to move on. On most slides, each press reveals one more idea.
SetsOperationsCountingOrdered pairsCartesian productRelations
1 · Sets
A set is a well-defined collection
Well-defined means we can always say whether an object belongs. “Stones weighing less than 1 kg” is a set. “Beautiful flowers” is not, because opinions differ.
Distinct elements. Repeats are counted once, so {2, 3, 3, 4} is written {2, 3, 4}.
Three ways to write a set
Descriptive: the set of even numbers less than 10
Tabular (roster): {2, 4, 6, 8}
Set builder: {x | x ∈ Z}, read “the set of all x such that x belongs to Z”
N naturalW wholeZ integersQ rational
2 · Kinds of sets
Empty, subset, disjoint, overlapping
Empty set { } or φ has no elements. {0} is not empty: it has one element, 0.
Subset. A ⊆ B when every element of A is also in B.
Disjoint sets have no common element, so A ∩ B = φ.
Overlapping sets share at least one element, and neither contains the other. For example {1, 3, 5} and {3, 5, 7}.
Universal set U. In a Venn diagram it is the rectangle, and other sets are closed shapes inside it.
3 · Operations
Union and intersection
A = {1, 2, 3, 4, 5} B = {1, 3, 4, 6, 7}
Union A ∪ B: everything in A or B or both.
{1, 2, 3, 4, 5, 6, 7}
{1, 2, 3, 4, 5, 6, 7}
Intersection A ∩ B: only what is in both.
{1, 3, 4}
{1, 3, 4}
4 · Operations
Difference and complement
Difference. A − B = {x | x ∈ A and x ∉ B}
If A = {1, 2, 3, 4, 5} and B = {3, 4, 6, 7, 8}, remove 3 and 4 from A: A − B = {1, 2, 5}.
If A = {1, 2, 3, 4, 5} and B = {3, 4, 6, 7, 8}, remove 3 and 4 from A: A − B = {1, 2, 5}.
Complement. A′ = U − A
If U = {1, …, 7} and A = {2, 4, 6}, then A′ = {1, 3, 5, 7}.
If U = {1, …, 7} and A = {2, 4, 6}, then A′ = {1, 3, 5, 7}.
Nothing is shared between A and its complement: A ∩ A′ = φ.
If A ⊂ B, then A − B = φ, because nothing is left after removing B’s elements.
5 · Laws of sets
Rules you can rely on
Same set twice. A ∪ A = A and A ∩ A = A
With the empty set. A ∪ φ = A and A ∩ φ = φ
When A ⊂ B. A ∩ B = A and A ∪ B = B
Distributive laws.
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Equal sets. If A − B = φ and B − A = φ, then A ⊆ B and B ⊆ A, so A = B.
6 · Counting
Do not count the overlap twice
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
The common elements are counted once in n(A) and again in n(B), so one copy is subtracted. For disjoint sets there is no overlap, so n(A ∪ B) = n(A) + n(B).
Class of 80. 40 like English, 34 like Maths, 9 like both.
n(E ∪ M) = 40 + 34 − 9 = 65, so 80 − 65 = 15 like neither.
n(E ∪ M) = 40 + 34 − 9 = 65, so 80 − 65 = 15 like neither.
Survey of 60. 25 watch channel A, 16 watch B, 4 watch both.
25 + 16 − 4 = 37 watch A or B.
25 + 16 − 4 = 37 watch A or B.
7 · Ordered pairs
Order matters
(a, b): a is the first element and b is the second element.
(2, 5) ≠ (5, 2). Swapping the elements gives a different pair, like two different points.
Equal pairs. (a, b) = (c, d) when a = c and b = d. So if (x, 3) = (2, y), then x = 2 and y = 3.
Example. Find x and y if (x − 3y, 5x + 1) = (4, 6).
From 5x + 1 = 6, x = 1. Then 1 − 3y = 4 gives y = −1.
From 5x + 1 = 6, x = 1. Then 1 − 3y = 4 gives y = −1.
8 · Cartesian product
Pair every element with every element
A × B = {(a, b) | a ∈ A, b ∈ B}
Example. A = {0, 1, 2}, B = {3, 4}
A × B = {(0,3), (0,4), (1,3), (1,4), (2,3), (2,4)}, which has 3 × 2 = 6 pairs.
A × B = {(0,3), (0,4), (1,3), (1,4), (2,3), (2,4)}, which has 3 × 2 = 6 pairs.
n(A × B) = n(A) × n(B), so n(A × A) = n2.
A × B ≠ B × A in general: same number of pairs, but written in the opposite order. They are equal only when A = B (non-empty sets).
If A = φ, no pair can be formed, so A × B = φ.
9 · Binary relations
A relation is a set of ordered pairs
A binary relation from A to B is any subset of A × B.
Domain = set of first elements. Range = set of second elements.
Example. R = {(2,1), (4,3), (2,2)}
Dom R = {2, 4} (the 2 is not repeated). Range = {1, 2, 3}.
Dom R = {2, 4} (the 2 is not repeated). Range = {1, 2, 3}.
Inverse. R−1 swaps every pair. If R = {(0,0), (2,4), (3,9)}, then R−1 = {(0,0), (4,2), (9,3)}.
Counting relations. A set with k elements has 2k subsets. With n(A) = 3 and n(B) = 2, A × B has 6 pairs, so there are 26 = 64 relations.
Void relation. R = φ, a relation with no ordered pairs.
10 · Common mistakes
Five traps to avoid
Calling {0} empty. It has one element. Only { } or φ is empty.
Adding n(A) + n(B) for overlapping sets. Subtract n(A ∩ B) once.
Swapping an ordered pair. (a, 1) ≠ (1, a).
Repeating elements in a domain. A set lists each element once.
Assuming A × B = B × A. It holds only when A and B are the same set.
11 · Quick check
Try these, then tap to check
12 · Your turn
Practise Unit 3
Three timed MCQ tests with an explanation after every answer.
Practice questions are independently written and are not official FBISE papers.