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CBSE · Class 11 · Mathematics

Operations on Sets (Ch 1: 1.9 and 1.10)

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

Union of sets

From daily life

A pizza shop counts everyone who ordered a pizza or a burger — people who ordered both are counted once.

Simple definition

The union A ∪ B is the set of all elements that are in A or in B or in both.

Formula: A ∪ B = {x : x ∈ A or x ∈ B}

AB

In the figure: The whole shaded region is A ∪ B.

Rule 1: Find a union — List all elements of A, then add the new ones from B.

  • Easy: {1, 2} ∪ {2, 3} = {1, 2, 3}.
  • Medium: {1, 2, 3, 4, 5, 6} ∪ {4, 5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8}.
  • Exam-type: A ∪ ∅ = A and A ∪ A = A.
Remember

A ⊆ A ∪ B and B ⊆ A ∪ B; also A ∪ B = B ∪ A.

Common mistake

Writing common elements twice.

Memory clue

UNION = bring everybody together (U for United).

Practice questions

Easy
  1. 1Find {1, 3, 5} ∪ {2, 3}.
  2. 2Find {a, b} ∪ ∅.
  3. 3If A ⊆ B, what is A ∪ B?
Medium
  1. 4A = {2, 4, 6}, B = {3, 6, 9}. Find A ∪ B.
  2. 5A = {x : x ∈ N, x < 4}, B = {x : x ∈ N, 2 < x < 6}. Find A ∪ B.
  3. 6Is A ∪ B = B ∪ A always?
Challenging
  1. 7Find P ∪ Q where P = multiples of 5 up to 30 and Q = multiples of 10 up to 30.
  2. 8If A ∪ B = A, what do you know?
Show answers (try them yourself first)
  1. 1{1, 2, 3, 5}
  2. 2{a, b}
  3. 3B
  4. 4{2, 3, 4, 6, 9}
  5. 5{1, 2, 3, 4, 5}
  6. 6Yes.
  7. 7{5, 10, 15, 20, 25, 30} (so P ∪ Q = P)
  8. 8B ⊆ A

Intersection of sets

From daily life

The students who play both cricket and chess are the ones counted in both lists.

Simple definition

The intersection A ∩ B is the set of elements that are in both A and B. If A ∩ B = ∅, the sets are disjoint.

Formula: A ∩ B = {x : x ∈ A and x ∈ B}

AB

In the figure: Only the overlap is shaded: A ∩ B.

Rule 1: Find an intersection — Pick the elements that appear in both lists.

  • Easy: {1, 2, 3} ∩ {3, 4} = {3}.
  • Medium: {1, 2, 3, 4, 5, 6} ∩ {4, 5, 6, 7, 8} = {4, 5, 6}.
  • Exam-type: {1, 3} ∩ {2, 4} = ∅, so the sets are disjoint.
Remember

A ∩ B ⊆ A and A ∩ B ⊆ B; A ∩ B = B ∩ A; A ∩ ∅ = ∅.

Common mistake

Confusing ∪ with ∩.

Memory clue

∩ looks like a cap that catches only the common part.

Spot the difference

A ∪ BA ∩ B
A or BA and B
Bigger or equal setSmaller or equal set

Practice questions

Easy
  1. 1Find {1, 2, 3, 4} ∩ {3, 4, 5}.
  2. 2Find {2, 4} ∩ ∅.
  3. 3If A ⊆ B, what is A ∩ B?
Medium
  1. 4A = multiples of 2 up to 20, B = multiples of 3 up to 20. Find A ∩ B.
  2. 5Are {1, 3, 5} and {2, 4, 6} disjoint?
  3. 6Find {x : x ∈ N, x < 8} ∩ {x : x ∈ N, x even}.
Challenging
  1. 7If A ∩ B = A, what do you know?
  2. 8Prime numbers up to 20 intersect odd numbers up to 20. How many elements?
Show answers (try them yourself first)
  1. 1{3, 4}
  2. 2∅
  3. 3A
  4. 4{6, 12, 18}
  5. 5Yes — nothing is common.
  6. 6{2, 4, 6}
  7. 7A ⊆ B
  8. 87 elements: {3, 5, 7, 11, 13, 17, 19}

Difference of sets

From daily life

Out of all students who play cricket, remove those who also play chess — the rest play only cricket.

Simple definition

The difference A − B is the set of elements that are in A but not in B. Note that A − B and B − A are usually different.

Formula: A − B = {x : x ∈ A and x ∉ B}

AB

In the figure: The shaded part of A, outside B, is A − B.

Rule 1: Find A − B — Strike off from A every element that is in B.

  • Easy: {1, 2, 3} − {3} = {1, 2}.
  • Medium: {1, 2, 3, 4, 5, 6} − {4, 5, 6, 7, 8} = {1, 2, 3} but {4, 5, 6, 7, 8} − {1, 2, 3, 4, 5, 6} = {7, 8}.
  • Exam-type: A − A = ∅ and A − ∅ = A.
Remember

A − B, A ∩ B and B − A never overlap, and together they make A ∪ B.

Common mistake

Assuming A − B = B − A.

Memory clue

A − B: A minus what B already took.

Practice questions

Easy
  1. 1Find {1, 2, 3, 4} − {2, 4}.
  2. 2Find {5, 6} − {1, 2}.
  3. 3What is A − A?
Medium
  1. 4A = {2, 4, 6, 8}, B = {4, 8, 12}. Find A − B and B − A.
  2. 5A = {x : x ∈ N, x ≤ 10}, B = even numbers up to 10. Find A − B.
  3. 6If A ∩ B = ∅, what is A − B?
Challenging
  1. 7Write A ∪ B using A − B, A ∩ B and B − A for A = {1, 2, 3}, B = {3, 4}.
  2. 8If A − B = ∅, what do you know?
Show answers (try them yourself first)
  1. 1{1, 3}
  2. 2{5, 6}
  3. 3∅
  4. 4{2, 6} and {12}
  5. 5{1, 3, 5, 7, 9}
  6. 6A
  7. 7{1, 2} ∪ {3} ∪ {4} = {1, 2, 3, 4}
  8. 8A ⊆ B

Complement of a set

From daily life

If U is the whole class, the students who are not in the football team form the "other" group.

Simple definition

The complement of A is the set of elements of U that are not in A: A′ = U − A.

Formula: A′ = {x : x ∈ U and x ∉ A}

AU

In the figure: The shaded region outside A (inside U) is A′.

Rule 1: Find A′ — List U, then remove the elements of A.

  • Easy: U = {1, 2, 3, 4, 5, 6}, A = {1, 2}: A′ = {3, 4, 5, 6}.
  • Medium: U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 4, 6, 8, 10}: A′ = {1, 3, 5, 7, 9}.
  • Exam-type: (A′)′ = A, U′ = ∅ and ∅′ = U.

Rule 2: De Morgan laws — Complement turns ∪ into ∩ and ∩ into ∪.

  • Easy: (A ∪ B)′ = A′ ∩ B′.
  • Medium: (A ∩ B)′ = A′ ∪ B′.
  • Exam-type: Check: U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3}, B = {3, 4}: (A ∪ B)′ = {5, 6} = A′ ∩ B′.
Remember

A ∪ A′ = U and A ∩ A′ = ∅.

Common mistake

Taking the complement without naming U — A′ depends on U.

Memory clue

Complement = the Left-outs of U.

Spot the difference

A − BA′
Both sets givenNeeds the universal set
Elements of A not in BElements of U not in A

Practice questions

Easy
  1. 1U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 3, 5}. Find A′.
  2. 2What is A ∪ A′?
  3. 3What is (A′)′?
Medium
  1. 4U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4}, B = {3, 4, 5, 6}. Find (A ∪ B)′.
  2. 5For the same sets find A′ ∩ B′.
  3. 6For the same sets find (A ∩ B)′.
Challenging
  1. 7If A′ = ∅, what is A?
  2. 8U = {x : x ∈ N, x ≤ 15}, A = multiples of 3. Find n(A′).
Show answers (try them yourself first)
  1. 1{2, 4, 6, 7, 8}
  2. 2U
  3. 3A
  4. 4{7, 8, 9, 10}
  5. 5{7, 8, 9, 10} — same as (A ∪ B)′
  6. 6{1, 2, 5, 6, 7, 8, 9, 10}
  7. 7A = U
  8. 8n(A) = 5, so n(A′) = 15 − 5 = 10

Counting with sets

From daily life

In a class of 40, some join music, some join dance, some join both. To know how many join at least one club, we must not count the "both" students twice.

Simple definition

For finite sets, the number of elements in the union is the sum of the two counts minus the common part.

Formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

AB

In the figure: The overlap is inside both circles, so it is counted twice in n(A) + n(B); subtract it once.

Rule 1: Find the union — Plug the three known numbers in.

  • Easy: n(A) = 20, n(B) = 15, n(A ∩ B) = 5: n(A ∪ B) = 30.
  • Medium: In a class of 50, 28 take Hindi, 30 take English, 10 take both: n(Hindi or English) = 48, so 2 take neither.
  • Exam-type: If A and B are disjoint, n(A ∪ B) = n(A) + n(B).

Rule 2: Find the common part — Rearrange: n(A ∩ B) = n(A) + n(B) − n(A ∪ B).

  • Easy: n(A) = 25, n(B) = 18, n(A ∪ B) = 35: n(A ∩ B) = 8.
  • Medium: Only A = n(A) − n(A ∩ B) = 25 − 8 = 17.
  • Exam-type: Only A (17) + only B (10) + both (8) = 35 = n(A ∪ B).
Remember

Always check "neither" using n(U) − n(A ∪ B).

Common mistake

Adding n(A) + n(B) and forgetting to subtract the overlap.

Memory clue

Add both, subtract the double-count.

Practice questions

Easy
  1. 1n(A) = 12, n(B) = 9, n(A ∩ B) = 4. Find n(A ∪ B).
  2. 2n(A) = 14, n(B) = 10 and A, B are disjoint. Find n(A ∪ B).
  3. 3n(A ∪ B) = 30, n(A) = 20, n(B) = 18. Find n(A ∩ B).
Medium
  1. 4In a group of 60, 35 like tea, 30 like coffee, 15 like both. How many like at least one?
  2. 5In the same group, how many like neither?
  3. 6In the same group, how many like only tea?
Challenging
  1. 7In a class of 45, 25 play cricket, 20 play kho-kho and 8 play neither. How many play both?
  2. 8If n(A) = 18, n(A ∩ B) = 7 and n(A ∪ B) = 31, find n(B).
Show answers (try them yourself first)
  1. 112 + 9 − 4 = 17
  2. 224
  3. 320 + 18 − 30 = 8
  4. 435 + 30 − 15 = 50
  5. 560 − 50 = 10
  6. 635 − 15 = 20
  7. 7Play at least one = 37; both = 25 + 20 − 37 = 8
  8. 8n(B) = 31 − 18 + 7 = 20