Lesson 11 of 11 · 15 min

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Sets from definitions to Venn diagrams
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Must-know facts

18 facts

1A set must be well-defined: membership is decided by fact, not opinion.
2In roster form order does not matter and repeated elements are written once.
3φ = { } has no element; {0} and {φ} each have exactly one element.
4φ is finite, n(φ) = 0, and φ is a subset of every set.
5Every set is a subset of itself: A ⊂ A.
6A = B if and only if A ⊂ B and B ⊂ A.
7∈ relates an element to a set; ⊂ relates a set to a set.
8A set with n elements has 2ⁿ subsets; {−1, 0, 1} has 8.
9N ⊂ Z ⊂ Q ⊂ R, and the irrationals T = R − Q.
10(a, b) excludes both ends, [a, b] includes both; all four intervals from a to b have length b − a.
11A bracket next to ∞ or −∞ is always round.
12A ∪ B: in A or B or both. A ∩ B: in both. Disjoint: A ∩ B = φ.
13If B ⊂ A then A ∪ B = A and A ∩ B = B.
14A − B = {x : x ∈ A, x ∉ B}; A − B ≠ B − A in general.
15A − B, A ∩ B and B − A are pairwise disjoint and their union is A ∪ B.
16A′ = U − A; A ∪ A′ = U, A ∩ A′ = φ, (A′)′ = A, φ′ = U, U′ = φ.
17De Morgan: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.
18A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

Common traps

Where marks are lost

Calling {φ} or {0} the empty set.

Count the elements inside the outer braces. {φ} holds one element (the empty set) and {0} holds one element (zero); only φ = { } holds none.

Writing {3, 4} ⊂ A for A = {1, 2, {3, 4}} because {3, 4} is "in" A.

{3, 4} is an element of A, so {3, 4} ∈ A. For ⊂, each of 3 and 4 would have to be an element of A, and neither is; {{3, 4}} ⊂ A is the true subset statement.

Counting a repeated letter or number twice when finding n(S).

Sets keep distinct elements only: the letters of CRICKET give 6 elements, not 7.

Writing [2, ∞] or [−∞, 5).

∞ is not a real number, so it can never be included; use [2, ∞) and (−∞, 5).

Assuming A − B = B − A, or answering A − B when B − A was asked.

Read the first set as the one you keep from: A − B keeps A's elements that are not in B.

Writing (A ∪ B)′ = A′ ∪ B′.

Complementing flips the operation: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′. Check with the four Venn regions if unsure.

Finding a complement without fixing the universal set.

A′ = U − A, so write U down first. The complement of the even numbers is the odd numbers in N, but in Z it also contains the negative odd numbers.

Forgetting φ and the set itself when listing or counting subsets.

A set of n elements has 2ⁿ subsets and 2ⁿ − 1 proper subsets; φ is always one of them.

Formulas

10 to know

Set-builder form

A = {x : x has property P}

Read: the set of all x such that x has property P.

Subset

A ⊂ B ⇔ (a ∈ A ⇒ a ∈ B)

φ ⊂ A and A ⊂ A for every set A.

Equal sets

A = B ⇔ A ⊂ B and B ⊂ A

The standard way to prove two sets equal.

Number of subsets

n(A) = n ⇒ number of subsets = 2ⁿ, proper subsets = 2ⁿ − 1

Counting result; φ and A itself are included in the 2ⁿ.

Intervals

(a, b) = {x : a < x < b}, [a, b] = {x : a ≤ x ≤ b}, [a, b) = {x : a ≤ x < b}, (a, b] = {x : a < x ≤ b}

Length b − a in every case; always a round bracket at ±∞.

Union and intersection

A ∪ B = {x : x ∈ A or x ∈ B}, A ∩ B = {x : x ∈ A and x ∈ B}

"or" includes elements in both.

Difference

A − B = {x : x ∈ A and x ∉ B} = A ∩ B′

Order matters.

Complement

A′ = {x : x ∈ U and x ∉ A} = U − A

A ∪ A′ = U, A ∩ A′ = φ, (A′)′ = A.

De Morgan's laws

(A ∪ B)′ = A′ ∩ B′, (A ∩ B)′ = A′ ∪ B′

Complement swaps ∪ and ∩.

Distributive law

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

∩ distributes over ∪.

Key terms

14 terms

Set: A well-defined collection of objects, where membership can be decided without opinion.
Roster form: Writing a set by listing its distinct elements inside braces.
Set-builder form: Writing a set by a property that its elements, and only they, satisfy.
Empty set: The set φ with no elements; it is finite and a subset of every set.
Finite set: A set that is empty or has a definite natural number of elements.
Equal sets: Sets with exactly the same elements.
Subset: A set all of whose elements belong to another set.
Proper subset: A subset that is not equal to the whole set.
Singleton: A set with exactly one element.
Interval: A set of all real numbers between two end points, each end included or not.
Universal set: The basic set of a discussion that contains every set being considered.
Venn diagram: A picture with a rectangle for U and closed curves for its subsets.
Disjoint sets: Sets with no element in common, so their intersection is φ.
Complement: The elements of the universal set that are not in the given set.