Sets ISC Class 11: Notes, Venn Diagrams and Solved Problems

What Sets is about

Weightage: Sets belongs to the unit Sets and Functions, which CISCE weights at 18 of the 80 marks in the ISC Class XI Mathematics Paper I (Theory). Class XI is examined by the school, so your school sets the paper within this weightage.Source: CISCE ISC Mathematics syllabus, examination year 2028, Class XI unit weightage

Key concepts

Sets and their representations

A set is a well-defined collection of distinct objects; 'the tall students of a class' is not a set (not well-defined), but 'the students taller than 160 cm' is. We write a ∈ A if a belongs to A and a ∉ A if not. Roster (tabular) form: list the elements in braces, without repetition; order does not matter. Set-builder form: {x : x has property P}, read 'the set of all x such that…'. Standard sets: N (natural numbers), Z (integers), Q (rationals), R (reals).

Worked example

Write {x : x ∈ N, x² < 30} in roster form, and {1/2, 2/3, 3/4, 4/5, 5/6, 6/7} in set-builder form.

Try it first, then show the solutionSolution

1. x² < 30 for x = 1, 2, 3, 4, 5 (since 6² = 36): {1, 2, 3, 4, 5}
2. Each element is n/(n + 1) for n = 1 to 6: {x : x = n/(n + 1), n ∈ N, n ≤ 6}

Answer: {1, 2, 3, 4, 5}; {x : x = n/(n + 1), n ∈ N, 1 ≤ n ≤ 6}

Empty, finite, infinite and equal sets

The empty (null, void) set ∅ = { } has no elements, e.g. {x : x ∈ R, x² + 1 = 0}. Note {0} and {∅} are not empty: each has one element. A set is finite if its elements can be counted to an end (n(A) is the number of elements, its cardinal number), otherwise infinite (N, the points on a line). A singleton set has one element. Equal sets A = B: exactly the same elements ({1, 2, 2} = {1, 2}). Equivalent sets: n(A) = n(B), even if the elements differ.

Worked example

Classify each set as empty, singleton, finite or infinite: A = {x : x ∈ N, 3 < x < 4}; B = {x : x is an even prime}; C = {x : x ∈ Z, x² ≤ 4}; D = {x : x is a multiple of 7}.

Try it first, then show the solutionSolution

1. A: no natural number lies strictly between 3 and 4: empty.
2. B = {2}: singleton. C = {−2, −1, 0, 1, 2}: finite, n(C) = 5.
3. D = {7, 14, 21, …} (with negatives and 0 if integers are meant): infinite.

Answer: A empty; B singleton; C finite (5 elements); D infinite

Subsets and the number of subsets

A is a subset of B (A ⊆ B, often written A ⊂ B) if every element of A is an element of B. Every set is a subset of itself, and ∅ is a subset of every set. A is a proper subset of B if A ⊂ B and A ≠ B. A = B exactly when A ⊂ B and B ⊂ A. If n(A) = n, A has $2^n$ subsets (each element is either in or out) and $2^n-1$ proper subsets. Do not confuse ∈ with ⊂: 1 ∈ {1, 2} but {1} ⊂ {1, 2}.

Worked example

List all subsets of A = {a, b, c}. How many are proper?

Try it first, then show the solutionSolution

1. ∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}
2. Total 2³ = 8; proper subsets exclude A itself: 7

Answer: 8 subsets, 7 proper

Intervals as subsets of R

For real numbers a < b: open interval (a, b) = {x : a < x < b}; closed interval [a, b] = {x : a ≤ x ≤ b}; half-open [a, b) = {x : a ≤ x < b} and (a, b] = {x : a < x ≤ b}. Infinite intervals: (a, ∞) = {x : x > a}, [a, ∞), (−∞, b), (−∞, b], and (−∞, ∞) = R. ∞ is never included (it is not a number), so it always takes a round bracket. The length of any of the four bounded intervals is b − a. On a number line, a filled dot marks an included end point and a hollow dot an excluded one.

Worked example

Write {x : x ∈ R, −2 < x ≤ 5} in interval notation and give its length. Write [3, ∞) in set-builder form.

Try it first, then show the solutionSolution

1. −2 excluded, 5 included: (−2, 5]; length = 5 − (−2) = 7
2. [3, ∞) = {x : x ∈ R, x ≥ 3}

Answer: (−2, 5], length 7; {x : x ∈ R, x ≥ 3}

Power set and universal set

The power set P(A) is the set of all subsets of A. If n(A) = n, then n(P(A)) = $2^n$. Its elements are sets: for A = {1, 2}, P(A) = {∅, {1}, {2}, {1, 2}}, so {1} ∈ P(A) and ∅ ∈ P(A). The universal set U is the set that contains all the sets under discussion as subsets; it depends on the context (for a problem about the digits of a number, U might be {0, 1, …, 9}). In a Venn diagram U is drawn as a rectangle.

Worked example

If A has 4 elements, how many elements does P(A) have, and how many does P(P(A)) have?

Try it first, then show the solutionSolution

1. n(P(A)) = 2⁴ = 16
2. n(P(P(A))) = 2¹⁶ = 65 536

Answer: 16; 65 536

Union, intersection, difference and complement

Union A ∪ B = {x : x ∈ A or x ∈ B}; intersection A ∩ B = {x : x ∈ A and x ∈ B}; A and B are disjoint if A ∩ B = ∅. Difference A − B = {x : x ∈ A, x ∉ B}; note A − B ≠ B − A in general. Complement A′ = U − A = {x ∈ U : x ∉ A}; and A − B = A ∩ B′. Symmetric difference (A − B) ∪ (B − A) contains elements in exactly one of the sets. Laws: union and intersection are commutative and associative; each distributes over the other: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C), A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C); A ∪ ∅ = A, A ∩ U = A; A ∪ A = A, A ∩ A = A.

Worked example

If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find A ∪ B, A ∩ B, A − B and B − A.

Try it first, then show the solutionSolution

1. A ∪ B = {1, 2, 3, 4, 5, 6}; A ∩ B = {3, 4}
2. A − B = {1, 2}; B − A = {5, 6}

Answer: {1, …, 6}; {3, 4}; {1, 2}; {5, 6}

Properties of complement and De Morgan's laws

Complement laws: A ∪ A′ = U; A ∩ A′ = ∅. Law of double complementation: (A′)′ = A. Laws of empty and universal sets: ∅′ = U and U′ = ∅. De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′: the complement of a union is the intersection of complements, and vice versa. To prove a set identity, show that each side is a subset of the other: take x in the left side and show it is in the right side, then reverse. Venn diagrams illustrate an identity but do not prove it.

Worked example

Prove De Morgan's law (A ∪ B)′ = A′ ∩ B′.

Try it first, then show the solutionSolution

1. Let x ∈ (A ∪ B)′. Then x ∉ A ∪ B, so x ∉ A and x ∉ B, i.e. x ∈ A′ and x ∈ B′, so x ∈ A′ ∩ B′. Hence (A ∪ B)′ ⊂ A′ ∩ B′.
2. Let x ∈ A′ ∩ B′. Then x ∉ A and x ∉ B, so x ∉ A ∪ B, i.e. x ∈ (A ∪ B)′. Hence A′ ∩ B′ ⊂ (A ∪ B)′.
3. Each side is a subset of the other, so they are equal.

Answer: Proved by showing each side is a subset of the other

Counting with Venn diagrams

For finite sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B) (the overlap was counted twice); if A and B are disjoint, n(A ∪ B) = n(A) + n(B). Also n(A − B) = n(A) − n(A ∩ B) ('only A'), and n(A′) = n(U) − n(A). Three sets: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C). In word problems, fill the Venn diagram from the centre outward: the triple overlap first, then the 'exactly two' regions, then the 'only one' regions, and finally 'none' = n(U) − n(A ∪ B ∪ C).

Worked example

In a class of 40 students, 25 study Mathematics, 20 study Physics and 10 study both. How many study neither? How many study only Physics?

Try it first, then show the solutionSolution

1. n(M ∪ P) = 25 + 20 − 10 = 35
2. Neither = 40 − 35 = 5; only Physics = 20 − 10 = 10

Answer: 5 study neither; 10 study only Physics

Maths in real life: Sets

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Set Operations, De Morgan's Laws and Venn Diagram Problems
• Subsets, Power Set and Intervals

Sets formula sheet

Number of subsets

$n(A)=n\Rightarrow 2^n\text{subsets},2^n-1\text{proper subsets}$

n(P(A)) = $2^n$.

Intervals

$(a,b),[a,b],[a,b),(a,b];\text{length}=b-a$

Round bracket: end point excluded.

Difference and complement

$A-B=A\cap B^{\prime },A^{\prime }=U-A$

A − B ≠ B − A in general.

Distributive laws

$A\cup (B\cap C)=(A\cup B)\cap (A\cup C);A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$

Each operation distributes over the other.

De Morgan's laws

$(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime },(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$

Swap ∪ and ∩ when taking complements.

Two sets

$n(A\cup B)=n(A)+n(B)-n(A\cap B)$

Only A: n(A) − n(A ∩ B).

Three sets

$n(A\cup B\cup C)=\sum n(A)-\sum n(A\cap B)+n(A\cap B\cap C)$

Fill Venn diagrams from the centre outward.

Sets: diagrams to practise

Practise each one on paper: draw it from the checklist, then tick off every label.

How the school exam asks Sets

The question patterns to expect, and the method that works for each.

Tricks to remember Sets

Short memory aids. Say each one aloud once, then use it on a question the same day.

• 2 to the n

  Each element says 'in' or 'out': 2 × 2 × … = 2ⁿ subsets.

  Use it when: Counting subsets and power sets.

• De Morgan flips

  Complement flips cup to cap and cap to cup: (A ∪ B)′ = A′ ∩ B′.

  Use it when: Applying De Morgan's laws.

• Round for infinity

  ∞ never gets a square bracket: it is not a number you can include.

  Use it when: Interval notation.

• Centre first

  In three-set problems, start with the triple overlap and work outward.

  Use it when: Venn-diagram surveys.

• Difference is 'and not'

  A − B = A ∩ B′: in A and not in B.

  Use it when: Simplifying expressions with differences.

Sets: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section D style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 5-mark question split into 1 + 1 + 1 + 2 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Sets

The slips examiners see most often, and what to write instead.

• ✗Mistake: Confusing ∈ and ⊂

  Students write 1 ⊂ {1, 2} or {1} ∈ {1, 2}.

  Wrong working

  {a} ∈ {a, b, c}.

  Right working

  a ∈ {a, b, c}; {a} ⊂ {a, b, c}; but {a} ∈ P({a, b, c}).

  ✓Do this: ∈ relates an element to a set; ⊂ relates a set to a set.

  Usually costs: 1 mark.

• ✗Mistake: Forgetting to subtract the triple overlap in 'exactly two'

  Students take n(A ∩ B) as 'only A and B'.

  Wrong working

  Exactly two = 40 + 50 + 90 = 180.

  Right working

  Subtract the 30 who speak all three from each: 10 + 20 + 60 = 90.

  ✓Do this: Fill the Venn diagram from the centre outward.

  Usually costs: 1-2 marks.

• ✗Mistake: Wrong brackets in interval answers

  Students include an end point that belongs to only one set, or use [ with ∞.

  Wrong working

  [−3, 5) ∩ (2, 8] = [2, 5].

  Right working

  (2, 5): 2 is not in B and 5 is not in A.

  ✓Do this: Check each end point against both sets.

  Usually costs: 1 mark.

• ✗Mistake: Proving identities with a Venn diagram alone

  A diagram illustrates one case; it is not a general proof.

  Wrong working

  Shading two diagrams and saying 'hence proved'.

  Right working

  Show each side is a subset of the other using an arbitrary element x.

  ✓Do this: Use the element method for every 'prove that' question.

  Usually costs: 2 marks.

• ✗Mistake: Treating {0} or {∅} as the empty set

  Students see 0 or ∅ inside braces and call the set empty.

  Wrong working

  {0} = ∅.

  Right working

  {0} has one element (0); {∅} has one element (∅); only { } is empty.

  ✓Do this: Count the elements, not their values.

  Usually costs: 1 mark.

Practice questions on Sets

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   Which of the following is the empty set?
   (a) {0}
   (b) {x : x ∈ R, x² + 1 = 0}
   (c) {∅}
   (d) {x : x ∈ N, x < 2}

   Hint

   Which equation has no real solution?

   Answer

   (b)

2. Q2 · 1 mark · MCQExam level

   (A ∪ B)′ is equal to:
   (a) A′ ∪ B′
   (b) A′ ∩ B′
   (c) A ∩ B
   (d) A − B

   Hint

   De Morgan.

   Answer

   (b)

3. Q3 · 1 mark · Very short answerExam level

   Write the interval [−1, 4) in set-builder form.

   Hint

   Square bracket includes.

   Answer

   {x : x ∈ R, −1 ≤ x < 4}

4. Q4 · 1 mark · Very short answerExam level

   Is {0} the empty set? Give a reason.

   Hint

   Count the elements.

   Answer

   No; it has one element, 0.

5. Q5 · 2 marks · Short answerExam level

   How many elements does the power set of {x, y, z} have? List those subsets that contain x.

   Hint

   2³.

   Answer

   8; {x}, {x, y}, {x, z}, {x, y, z}.

6. Q6 · 2 marks · Short answerHarder exam level

   If A = {x : x is a prime less than 20} and B = {x : x is an odd natural number less than 20}, find A − B and B − A.

   Hint

   List both sets.

   Answer

   A = {2, 3, 5, 7, 11, 13, 17, 19}; B = {1, 3, 5, …, 19}. A − B = {2}; B − A = {1, 9, 15}.

7. Q7 · 2 marks · Short answerHarder exam level

   If n(A) = 3 and n(B) = 5, find the least and greatest possible values of n(A ∪ B).

   Hint

   Overlap from 0 to 3.

   Answer

   Least 5 (A ⊂ B); greatest 8 (A and B disjoint).

8. Q8 · 2 marks · Short answerHarder exam level

   Prove that A ∪ (A ∩ B) = A.

   Hint

   A ∩ B ⊂ A.

   Answer

   A ⊂ A ∪ (A ∩ B) always. Conversely, if x ∈ A ∪ (A ∩ B), then x ∈ A or x ∈ A ∩ B ⊂ A, so x ∈ A. Hence equal.

9. Q9 · 2 marks · Short answerHarder exam level

   If A ∩ B = ∅, show that A ⊂ B′.

   Hint

   Take x ∈ A.

   Answer

   Let x ∈ A. Since A ∩ B = ∅, x ∉ B, so x ∈ B′. Hence A ⊂ B′.

10. Q10 · 3 marks · Short answerHarder exam level

    In a group of 70 people, 37 like coffee and 52 like tea, and every person likes at least one of the two drinks. How many like both? How many like only tea?

    Hint

    n(C ∪ T) = 70.

    Answer

    Both = 37 + 52 − 70 = 19; only tea = 52 − 19 = 33 (only coffee 18).

11. Q11 · 3 marks · Short answerHarder exam level

    Let U = {1, 2, …, 12}, A = {x : x is a multiple of 3} and B = {x : x is even}, both subsets of U. Verify that (A ∩ B)′ = A′ ∪ B′.

    Hint

    List A and B first, then find A ∩ B.

    Answer

    A = {3, 6, 9, 12}, B = {2, 4, 6, 8, 10, 12}, A ∩ B = {6, 12}, so (A ∩ B)′ = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}. A′ ∪ B′ = {1, 2, 4, 5, 7, 8, 10, 11} ∪ {1, 3, 5, 7, 9, 11} = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}. Equal.

12. Q12 · 5 marks · Long answerHarder exam level

    (a) State and prove the distributive law A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). [3]
    (b) Illustrate it with a Venn diagram. [1]
    (c) If n(A) = 10, n(B) = 15 and n(A ∩ B) = 4, find n(A − B) and n(A ∪ B). [1]

    Hint

    Element method.

    Answer

    (a) x ∈ A ∩ (B ∪ C) ⇔ x ∈ A and (x ∈ B or x ∈ C) ⇔ (x ∈ A and x ∈ B) or (x ∈ A and x ∈ C) ⇔ x ∈ (A ∩ B) ∪ (A ∩ C). (b) Shade A ∩ (B ∪ C) and (A ∩ B) ∪ (A ∩ C): the same region. (c) n(A − B) = 6; n(A ∪ B) = 21.

13. Q13 · 3 marks · Short answerHOTS / competency

    Prove that (A − B) ∪ (B − A) = (A ∪ B) − (A ∩ B).

    Hint

    Both sides are 'in exactly one'.

    Answer

    x ∈ LHS ⇔ (x ∈ A, x ∉ B) or (x ∈ B, x ∉ A) ⇔ x is in exactly one of A, B ⇔ x ∈ A ∪ B and x ∉ A ∩ B ⇔ x ∈ RHS.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.Number of subsets of a set with n elements?Show answer

   2ⁿ.

2. 2.Is ∅ a subset of every set?Show answer

   Yes.

3. 3.Write (a, b] in set-builder form.Show answer

   {x : a < x ≤ b}.

4. 4.A − B in terms of complement?Show answer

   A ∩ B′.

5. 5.State De Morgan's laws.Show answer

   (A ∪ B)′ = A′ ∩ B′; (A ∩ B)′ = A′ ∪ B′.

6. 6.n(A ∪ B) formula?Show answer

   n(A) + n(B) − n(A ∩ B).

7. 7.What is P(A)?Show answer

   The set of all subsets of A.

8. 8.When are two sets disjoint?Show answer

   When their intersection is empty.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you convert a set between roster and set-builder form?
• ☐Can you tell ∅, {0} and {∅} apart?
• ☐Can you count and list subsets?
• ☐Can you write intervals correctly with brackets?
• ☐Can you find unions, intersections and differences of intervals?
• ☐Can you prove De Morgan's laws by the element method?
• ☐Can you use n(A ∪ B) = n(A) + n(B) − n(A ∩ B)?
• ☐Can you solve a three-set survey problem?

Sets: frequently asked questions