Lesson 11 of 11 · 15 min
Chapter review
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Sets from definitions to Venn diagrams
Concepts, NCERT questions and extra practice
Must-know facts
18 facts
- 01A set must be well-defined: membership is decided by fact, not opinion.
- 02In roster form order does not matter and repeated elements are written once.
- 03φ = { } has no element; {0} and {φ} each have exactly one element.
- 04φ is finite, n(φ) = 0, and φ is a subset of every set.
- 05Every set is a subset of itself: A ⊂ A.
- 06A = B if and only if A ⊂ B and B ⊂ A.
- 07∈ relates an element to a set; ⊂ relates a set to a set.
- 08A set with n elements has 2ⁿ subsets; {−1, 0, 1} has 8.
- 09N ⊂ 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
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.