Sets

In this chapter: sets and their representation, empty, finite and infinite sets, equal sets, subsets and power set, intervals, universal set and Venn diagrams, union, intersection, difference, complement, De Morgan's laws and counting with n(A ∪ B).

What counts as a set

A set is a well-defined collection of objects. Well-defined means that for any object you can decide, without argument, whether it belongs to the collection or not. "The vowels of the English alphabet" is a set. "The five best batsmen in India" is not, because two people will give two different lists.

The objects are called elements or members. We write a ∈ A for "a belongs to A" and b ∉ A for "b does not belong to A". Sets are named with capital letters and elements with small letters.

Roster form and set-builder form

In roster form you list the elements inside braces, separated by commas: the set of prime numbers less than 12 is {2, 3, 5, 7, 11}. The order of listing does not matter, and an element is never written twice. The letters of the word SCHOOL form the set {S, C, H, O, L}.

In set-builder form you describe a property shared by all elements and by nothing else: {x : x is a natural number and x² < 30}, which in roster form is {1, 2, 3, 4, 5}. The colon is read "such that". Set-builder form is the only practical choice for sets like {x : x ∈ R, 0 < x < 1}, which cannot be listed.

Types of sets

• Empty (null, void) set, written φ or { }: it has no element. Example: {x : x ∈ N, 1 < x < 2}, or the set of real x with x² + 1 = 0.
• Singleton set: exactly one element, such as {0}. Note that {0} is not empty.
• Finite set: empty, or with a definite number of elements. n(A) denotes the number of elements, also called the cardinal number. For A = {a, e, i, o, u}, n(A) = 5.
• Infinite set: not finite, like N or the set of points on a line. Some infinite sets can still be written in roster form, such as N = {1, 2, 3, ...}; others, such as R, cannot.
• Equal sets: A = B when they have exactly the same elements. {1, 2, 2, 3} and {3, 1, 2} are equal because repetition and order do not matter.

Subsets and power set

A is a subset of B, written A ⊂ B, if every element of A is also an element of B. If A ⊂ B and A ≠ B, A is a proper subset of B and B is a superset of A. A few facts follow straight from the definition:

• Every set is a subset of itself: A ⊂ A.
• The empty set is a subset of every set, since it has no element that could fail the test.
• A = B exactly when A ⊂ B and B ⊂ A. This is how equality of sets is proved.

The collection of all subsets of A is the power set P(A). For A = {1, 2}, P(A) = {φ, {1}, {2}, {1, 2}}. Each element either goes into a subset or stays out, which gives the counting rule:

Also keep the chain N ⊂ Z ⊂ Q ⊂ R in mind, with the irrationals T = {x : x ∈ R and x ∉ Q}, so that Q and T together make up R.

Intervals as subsets of R

For real numbers a < b, the following subsets of R appear in every domain and range question you will solve.

In each case the length of the interval is b − a. Infinity is never included, so write (−∞, 5] or (2, ∞), with a round bracket next to ∞.

Universal set and Venn diagrams

The universal set U is the basic set of which all sets in a given discussion are subsets. When you study triangles, U might be the set of all triangles; for a quadratic equation, U might be R. In a Venn diagram U is drawn as a rectangle and its subsets as circles inside it.

Operations on sets

If A ∩ B = φ, A and B are disjoint. The "or" in the union is inclusive: an element in both sets is counted once. Note that A − B is also A ∩ B′, and that A − B and B − A are different sets in general.

Laws you should be able to quote

For complements: A ∪ A′ = U, A ∩ A′ = φ, (A′)′ = A, φ′ = U and U′ = φ.

An easy way to remember De Morgan: taking the complement flips every ∪ to ∩ and every ∩ to ∪, and puts a dash on each set.

Worked example: U = {1, 2, 3, ..., 10}, A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}. Find A ∪ B, A ∩ B, A − B, B − A and verify (A ∪ B)′ = A′ ∩ B′.
Solution: A ∪ B = {1, 2, 3, 4, 5, 6, 8}; A ∩ B = {2, 4, 6}; A − B = {1, 3, 5}; B − A = {8}. Then (A ∪ B)′ = {7, 9, 10}. Separately, A′ = {7, 8, 9, 10} and B′ = {1, 3, 5, 7, 9, 10}, so A′ ∩ B′ = {7, 9, 10}. Both sides agree.

Counting elements

For finite sets, adding n(A) and n(B) counts the common elements twice, so you subtract them once:

Worked example: In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two. How many like both? How many like coffee but not tea?
Solution: Let C and T be the two sets. Everyone likes at least one, so n(C ∪ T) = 70. Then 70 = 37 + 52 − n(C ∩ T), which gives n(C ∩ T) = 19. Coffee but not tea: n(C − T) = 37 − 19 = 18.

When the problem is wordy, draw the Venn diagram and fill the innermost region first, then work outwards. This avoids nearly every counting error.

Common mistakes: (1) φ and {φ} are different: {φ} has one element, so it is not empty. (2) If a ∈ A, then {a} ⊂ A; writing {a} ∈ A is wrong unless {a} itself is listed as an element. (3) The number of proper subsets 2^m − 1 includes φ; if the question says non-empty proper subsets, the answer is 2^m − 2. (4) Repeating elements in roster form, for example writing {S, C, H, O, O, L}. (5) Treating A − B and B − A as the same set. (6) Putting a square bracket next to ∞ in an interval.

Exam focus

• Switching between roster and set-builder forms, and spotting which given collection is empty or singleton.
• Counting subsets, proper subsets and elements of the power set.
• Writing solution sets of inequalities in interval notation; this carries straight into domain and range.
• Using De Morgan's laws and the distributive laws to simplify set expressions.
• Two-set and three-set counting problems with n(A ∪ B) and n(A ∪ B ∪ C).

Practice questions