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State Board · Class 9 · Mathematics Part 1

Lesson 1: Sets

5 concepts · 40 practice questions with answers · Free to read

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Sets and methods of writing sets

From daily life

"Months of the year with 30 days" is a clear list: everyone agrees on it. But "tasty fruits" is not clear, because tastes differ. Mathematics only uses collections that are clear.

Simple definition

A set (संच) is a well-defined collection of objects; each object is an element (घटक).

Rule 1: Notation — A set is named with a capital letter (A, B, C ...) and its elements with small letters. a ∈ A means a belongs to A; a ∉ A means it does not. N = {1, 2, 3, ...} natural numbers; W = {0, 1, 2, 3, ...} whole numbers; I = {..., −2, −1, 0, 1, 2, ...} integers; Q rational numbers; R real numbers.

  • Easy : Easy: 5 ∈ N, but −5 ∉ N.
  • Medium : Medium: 0 ∈ W but 0 ∉ N.
  • Exam-type : Exam: Is "good singers in class" a set? No, "good" is relative. "Students born in March" is a set.

Rule 2: Two ways to write a set — Listing (roster) method (यादी पद्धत): write all elements once inside curly brackets; the order does not matter. Letters of "BOOK" give {b, o, k}. Rule (set builder) method (गुणधर्म पद्धत): write the general element and its property. A = {x : x ∈ N, x < 6} = {1, 2, 3, 4, 5}. B = {x : x is an even number between 1 and 11} = {2, 4, 6, 8, 10}.

The set of even natural numbers below 10 written by listing and by ruleTwo ways to write the same setListing (roster) methodA = {2, 4, 6, 8}Rule (set-builder) methodA = {x : x is an evennatural number, x < 10}

In the figure : In the figure a set can be written by listing its elements or by stating a rule.

  • Easy : Easy: Days of a weekend = {Saturday, Sunday}.
  • Medium : Medium: {x : x ∈ N, x is a multiple of 3, x < 13} = {3, 6, 9, 12}.
  • Exam-type : Exam: Write {5, 10, 15, 20} by rule: {x : x = 5n, n ∈ N, n ≤ 4}.
Remember

Write each element once. Order does not matter. Clear collections only.

Common mistake

Repeating an element, as {b, o, o, k}. List it once: {b, o, k}.

Shortcut

Memory clue: "संच = स्पष्ट संग्रह; घटक एकदाच".

Spot the difference

MethodExample
Listing{2, 4, 6, 8, 10}
Rule{x : x is an even number between 1 and 11}
Symbols5 ∈ N, −5 ∉ N

Practice questions

Easy
  1. 1What is a set?
  2. 2What does a ∈ A mean?
  3. 3Is "tall boys" a set?
Medium
  1. 4Write the set of letters in "MAMA".
  2. 5Which is smaller: N or W? Why?
  3. 6Write {x : x ∈ N, x < 5} in listing form.
Challenging
  1. 7Write {x : x is a prime number less than 12}.
  2. 8Does the order {3, 1, 2} vs {1, 2, 3} change the set?
Show answers (try them yourself first)
  1. 1A well-defined collection of objects.
  2. 2a is an element of A.
  3. 3No, "tall" is relative.
  4. 4{m, a}.
  5. 5N, since W also has 0.
  6. 6{1, 2, 3, 4}.
  7. 7{2, 3, 5, 7, 11}.
  8. 8No, same set.

Types of sets and equal sets

From daily life

A bag with one pen is a "one-item" bag. An empty bag has nothing. A bag of marbles can be counted, but the points on a line cannot. Sets come in the same kinds.

Simple definition

Sets are classified as singleton, empty, finite and infinite; two sets are equal if they have exactly the same elements.

Rule 1: Types of sets — Singleton: exactly one element, {7}. Empty (null): no element, written { } or φ, e.g. natural numbers between 5 and 6. Finite: a null set or a set with a limited, countable number of elements, e.g. days of a week. Infinite: unlimited elements, e.g. N. N, W, I, Q and R are all infinite.

Singleton {7}, empty set, finite set {3, 6, 9, 12} and infinite set of natural numbersTypes of setsSingleton{7}Empty set{ } or φFinite{3, 6, 9, 12}InfiniteN = {1, 2, 3, …}

In the figure : In the figure sets can have one, none, a limited number or unlimited elements.

  • Easy : Easy: {9} is a singleton set.
  • Medium : Medium: {x : x ∈ N, 3 < x < 4} is the empty set.
  • Exam-type : Exam: Is {x : x ∈ N, x < 100} finite or infinite? Finite, it has 99 elements.

Rule 2: Equal sets — A = B if every element of A is in B and every element of B is in A. The order does not matter. {m, a, t, h} = {h, a, t, m}. {2, 4, 6} ≠ {2, 4, 8} because 6 is in the first but not in the second.

  • Easy : Easy: {1, 2, 3} = {3, 2, 1}.
  • Medium : Medium: {a, a, b} = {a, b} since repeats are ignored.
  • Exam-type : Exam: {x : x is a letter of "LEVEL"} = {l, e, v}, which has 3 elements.
Remember

Empty set φ has no element. φ is not {0}. {0} has one element.

Common mistake

Writing the empty set as {φ}. That has one element. Write φ or { }.

Shortcut

Memory clue: "घटक सारखे = संच समान, क्रम काहीही".

Spot the difference

TypeExample
Singleton{7}
Emptyφ
Finite{2, 4, 6, 8}
InfiniteN

Practice questions

Easy
  1. 1What is a singleton set?
  2. 2Symbol for the empty set?
  3. 3Is N finite?
Medium
  1. 4Is {x : x ∈ N, 8 < x < 9} empty?
  2. 5Is {1, 2} = {2, 1}?
  3. 6Is φ = {0}?
Challenging
  1. 7How many elements in {x : x is a letter of "BANANA"}?
  2. 8Are {x : x ∈ N, x < 4} and {1, 2, 3} equal?
Show answers (try them yourself first)
  1. 1A set with exactly one element.
  2. 2φ or { }.
  3. 3No, infinite.
  4. 4Yes.
  5. 5Yes.
  6. 6No, {0} has one element.
  7. 73 ({b, a, n}).
  8. 8Yes, both have the same elements.

Venn diagrams, subsets, universal set and complement

From daily life

A small box of pens sits inside a big school bag. The bag holds everything we are talking about, and the box is a part of it. A Venn diagram draws this picture: a rectangle for the bag, circles for the parts.

Simple definition

B is a subset (उपसंच) of A if every element of B is in A. The universal set (सार्वत्रिक संच) U holds all the sets we study. The complement (पूरक संच) B′ is what is in U but not in B.

Rule 1: Subsets and Venn diagrams — John Venn first used closed figures to show sets. B ⊆ A means B is a subset of A. Every set is a subset of itself (A ⊆ A). The empty set is a subset of every set (φ ⊆ A). If A ⊆ B and B ⊆ A then A = B. Example: {2, 4} ⊆ {2, 4, 6}.

Left: B drawn inside A. Right: the shaded part outside A is the complementSubset and complementUABUAA′B ⊆ A:all of B is inside AA′ = in U but notin A (shaded)

In the figure : In the figure Venn diagrams show subsets and complements at a glance.

  • Easy : Easy: {1} ⊆ {1, 2}.
  • Medium : Medium: φ ⊆ {a, b, c}.
  • Exam-type : Exam: Is {1, 5} ⊆ {1, 2, 3, 4}? No, 5 is missing from the second set.

Rule 2: Universal set and complement — B′ = {x : x ∈ U and x ∉ B}. Let U = {1, 2, 3, ..., 10} and A = {2, 4, 6, 8, 10}; then A′ = {1, 3, 5, 7, 9}. Also (A′)′ = A, A and A′ have no common element, U′ = φ and φ′ = U. The universal set is shown by a rectangle.

  • Easy : Easy: U = {1, 2, 3}, A = {1}; A′ = {2, 3}.
  • Medium : Medium: U = {1, 2, ..., 6}, A = {1, 2, 3}; A′ = {4, 5, 6}.
  • Exam-type : Exam: U = {x : x ∈ N, x ≤ 12}, B = {3, 6, 9, 12}. Then B′ = {1, 2, 4, 5, 7, 8, 10, 11}, which has 12 − 4 = 8 elements.
Remember

Every set is a subset of itself. φ is a subset of every set. B′ lives inside U.

Common mistake

Forgetting U when finding the complement. Always name U first.

Shortcut

Memory clue: "पूरक = U मधले, पण त्यात नसलेले".

Spot the difference

IdeaSymbolExample
SubsetB ⊆ A{2, 4} ⊆ {2, 4, 6}
Universal setU{1, 2, ..., 10}
ComplementA′{1, 3, 5, 7, 9}

Practice questions

Easy
  1. 1Who first used closed figures for sets?
  2. 2What does B ⊆ A mean?
  3. 3Is φ a subset of every set?
Medium
  1. 4What shape shows U in a Venn diagram?
  2. 5U = {1, 2, 3, 4}, A = {1, 2}. Find A′.
  3. 6What is (A′)′?
Challenging
  1. 7What is U′?
  2. 8If A ⊆ B and B ⊆ A, what can we say?
Show answers (try them yourself first)
  1. 1John Venn.
  2. 2Every element of B is in A.
  3. 3Yes.
  4. 4A rectangle.
  5. 5{3, 4}.
  6. 6A.
  7. 7φ.
  8. 8A = B.

Intersection, disjoint sets and union

From daily life

Two friends' music playlists may share some songs. The shared songs are the intersection; all songs from both lists together are the union.

Simple definition

Intersection (छेदसंच) A ∩ B holds the common elements; union (संयोगसंच) A ∪ B holds the elements in A or B or both. Sets with nothing in common are disjoint (विभक्त).

Rule 1: Intersection and disjoint sets — A ∩ B = {x : x ∈ A and x ∈ B}. Let A = {1, 2, 3, 4, 5, 6} and B = {4, 5, 6, 7, 8}. Then A ∩ B = {4, 5, 6}. A ∩ B = B ∩ A; A ∩ A = A; A ∩ φ = φ; A ∩ A′ = φ; if A ⊆ B then A ∩ B = A. {10, 20, 30} and {15, 25} are disjoint, so their intersection is φ.

Three Venn diagrams: intersection shaded in the overlap, union shaded in both circles, and two separate circles for disjoint setsA = {1, 2, 3, 4} B = {3, 4, 5, 6}UAB3, 4UABUABA ∩ B = {3, 4}commonA ∪ B = {1…6}everythingDisjoint:nothing common

In the figure : In the figure intersection is the overlap and union is everything in either circle.

  • Easy : Easy: {1, 2} ∩ {2, 3} = {2}.
  • Medium : Medium: {1, 3, 5} ∩ {2, 4, 6} = φ, so they are disjoint.
  • Exam-type : Exam: Factors of 12 are A = {1, 2, 3, 4, 6, 12}; factors of 18 are B = {1, 2, 3, 6, 9, 18}. A ∩ B = {1, 2, 3, 6}.

Rule 2: Union — A ∪ B = {x : x ∈ A or x ∈ B}. With A = {1, 2, 3, 4, 5, 6} and B = {4, 5, 6, 7, 8}: A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}. A ∪ B = B ∪ A; A ∪ A = A; A ∪ φ = A; A ∪ A′ = U; if A ⊆ B then A ∪ B = B. Write common elements only once.

Live figure

Intersection, disjoint sets and union

In the app you can move this figure with your finger.

Open in the app
  • Easy : Easy: {1, 2} ∪ {2, 3} = {1, 2, 3}.
  • Medium : Medium: {a, b} ∪ φ = {a, b}.
  • Exam-type : Exam: A ∪ B for the factors of 12 and 18 is {1, 2, 3, 4, 6, 9, 12, 18}, which has 8 elements.
Remember

∩ means "and" (common). ∪ means "or" (everything).

Common mistake

Writing common elements twice in a union. List them once.

Shortcut

Memory clue: "∩ = समान; ∪ = सगळे".

Spot the difference

OperationMeaningExample
A ∩ BCommon{4, 5, 6}
A ∪ BAll{1, 2, ..., 8}
DisjointA ∩ B = φ{10, 20, 30} and {15, 25}

Practice questions

Easy
  1. 1What is A ∩ B?
  2. 2What is A ∪ B?
  3. 3What are disjoint sets?
Medium
  1. 4A = {1, 2, 3}, B = {3, 4}. Find A ∩ B.
  2. 5Find A ∪ B for the same sets.
  3. 6What is A ∩ A′?
Challenging
  1. 7What is A ∪ A′?
  2. 8If A ⊆ B, what are A ∩ B and A ∪ B?
Show answers (try them yourself first)
  1. 1The common elements of A and B.
  2. 2All elements in A or B or both.
  3. 3Sets with no common element.
  4. 4{3}.
  5. 5{1, 2, 3, 4}.
  6. 6φ.
  7. 7U.
  8. 8A ∩ B = A and A ∪ B = B.

Number of elements in a set

From daily life

If 20 children like tea and 15 like coffee, the group is not 35 children when some like both. We must remove the doubly counted children once.

Simple definition

n(A) is the number of elements (घटकसंख्या) of A. The key formula is n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

Rule 1: The formula — In n(A) + n(B) the common elements are counted twice, so subtract them once: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Take A = {2, 4, 6, 8, 10} and B = {4, 8, 12, 16}. A ∩ B = {4, 8}, so n(A ∪ B) = 5 + 4 − 2 = 7. If A and B are disjoint, n(A ∪ B) = n(A) + n(B).

A Venn diagram with 16 playing only football, 8 both and 10 only chess; 6 play neitherA class of 40 studentsU16810footballonlybothchessonlyFootball 24Chess 18neither 6n(A ∪ B) = 24 + 18 − 8 = 34 (40 − 34 = 6 play neither)

In the figure : In the figure subtracting the overlap once stops it being counted twice.

  • Easy : Easy: n(A) = 6, n(B) = 5, disjoint, so n(A ∪ B) = 11.
  • Medium : Medium: n(A) = 12, n(B) = 9, n(A ∩ B) = 4, so n(A ∪ B) = 17.
  • Exam-type : Exam: n(A ∪ B) = 50, n(A) = 30, n(B) = 28. Then n(A ∩ B) = 30 + 28 − 50 = 8.

Rule 2: Word problems — In a group of 60, 35 like tea (T) and 40 like coffee (C), and all like at least one. Both = 35 + 40 − 60 = 15. Only tea = 35 − 15 = 20; only coffee = 40 − 15 = 25. Check: 20 + 15 + 25 = 60. If some like neither, first subtract them: 80 students, 10 like neither, so n(M ∪ E) = 70; with n(M) = 50, n(E) = 45, both = 50 + 45 − 70 = 25.

  • Easy : Easy: 10 like only A, 5 like both, 3 like only B: total 18.
  • Medium : Medium: n(U) = 40, n(A ∪ B) = 33, so 40 − 33 = 7 like neither.
  • Exam-type : Exam: In the example with 80 students, only Marathi = 50 − 25 = 25 and only English = 45 − 25 = 20. Check 25 + 25 + 20 + 10 = 80.
Remember

Subtract the overlap once. Do not forget the "neither" group.

Common mistake

Writing n(A ∪ B) = n(A) + n(B) always. That is only for disjoint sets.

Shortcut

Memory clue: "जोड दोघांना, वजा सामाईक एकदा".

Spot the difference

CaseFormula
Generaln(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Disjointn(A ∪ B) = n(A) + n(B)
Bothn(A ∩ B) = n(A) + n(B) − n(A ∪ B)

Practice questions

Easy
  1. 1What does n(A) mean?
  2. 2State the formula for n(A ∪ B).
  3. 3Why subtract n(A ∩ B)?
Medium
  1. 4n(A) = 7, n(B) = 8, disjoint. Find n(A ∪ B).
  2. 5n(A) = 10, n(B) = 6, n(A ∩ B) = 3. Find n(A ∪ B).
  3. 6n(A ∪ B) = 40, n(A) = 25, n(B) = 22. Find n(A ∩ B).
Challenging
  1. 7In a class of 50, 30 play kabaddi and 25 play kho-kho, all play at least one. How many play both?
  2. 8In the class above, how many play only kabaddi?
Show answers (try them yourself first)
  1. 1Number of elements in A.
  2. 2n(A) + n(B) − n(A ∩ B).
  3. 3Common elements are counted twice.
  4. 415.
  5. 513.
  6. 67.
  7. 730 + 25 − 50 = 5.
  8. 830 − 5 = 25.

Quick revision

  • A ∩ B = common elements; A ∪ B = all elements of A or B; A′ = elements of U not in A; (A′)′ = A.
  • B ⊆ A when every element of B is in A; the empty set is a subset of every set.
  • n(A ∪ B) = n(A) + n(B) − n(A ∩ B).