Tag: universal set

Sets: Union, Intersection and Solving Two-Set Venn Diagram Problems

Aven Brisail2 Comments

FORM 2 • TERM 1 • LESSON 13

In Form 1 you met sets, subsets and Venn diagrams and solved some Venn diagram word problems. This lesson pulls those ideas together and goes further: you will read diagrams of subsets, disjoint and intersecting sets, count the members of a union and an intersection, and solve two-set problems – including ones where you have to find an unknown.

Learning outcomes

By the end of this lesson you should be able to:

  • 2.2.1.1 – Interpret information about subsets, disjoint sets and intersecting sets.
  • 2.2.1.2 – Count the members of the union and the intersection of two sets.
  • 2.2.1.3 – Show information about sets on the right Venn diagram.
  • 2.2.1.4 – Count the members of the union of two sets, both intersecting and disjoint.
  • 2.2.1.5 – Solve two-set Venn diagram problems when you are given the number of members in the sets.
  • 2.2.1.6 – Use a Venn diagram to solve word problems involving two sets.

Quick reminder

  • The universal set U contains everything being talked about. It is drawn as a rectangle.
  • n(A) means the number of members of set A.
  • The intersection A ∩ B is the set of members in both A and B (the overlap).
  • The union A ∪ B is the set of members in A or B or both (everything inside the circles).
  • The complement A′ is everything in U that is not in A.

Subsets, disjoint sets and intersecting sets

Three Venn diagrams: B inside A, two circles that do not touch, and two overlapping circles UB ⊂ A (subset) AB UDisjoint AB UIntersecting AB
Figure 1 – The three ways two sets can sit together.
  • Subset: every member of B is also in A, so B ⊂ A and B’s circle is drawn inside A’s. Then A ∩ B = B and A ∪ B = A.
  • Disjoint: A and B have no members in common, so A ∩ B = { } (the empty set) and the circles do not touch.
  • Intersecting: A and B share some members but not all, so the circles overlap.

Counting a union and an intersection

If A = {1, 2, 3, 4, 6, 12} (the factors of 12) and B = {1, 2, 3, 6, 9, 18} (the factors of 18):

  • A ∩ B = {1, 2, 3, 6}, so n(A ∩ B) = 4.
  • A ∪ B = {1, 2, 3, 4, 6, 9, 12, 18}, so n(A ∪ B) = 8.

Notice that n(A) + n(B) = 6 + 6 = 12, but the union has only 8 members. That is because the 4 shared members were counted twice. Taking them away once gives the key formula:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

For disjoint sets there is no overlap, so n(A ∪ B) = n(A) + n(B).

Filling in a Venn diagram from numbers

  1. Put the number in the intersection in first.
  2. Work out “A only” = n(A) − n(A ∩ B) and “B only” = n(B) − n(A ∩ B).
  3. Work out the region outside both circles = n(U) − n(A ∪ B).
  4. Check that all four regions add up to n(U).

TRY ITFill in a two-set Venn diagram

Type the four facts that two-set problems usually give you. The diagram fills in each region and the steps show the working.

Venn diagram with two overlapping circles A and B inside a rectangle U U AB

Worked examples

Example 1: Reading a diagram

Question: In a Venn diagram, the regions are: A only 9, A ∩ B 4, B only 6, outside both 3. Find n(A), n(B), n(A ∪ B), n(U) and n(A′).

  1. n(A) = 9 + 4 = 13 and n(B) = 4 + 6 = 10.
  2. n(A ∪ B) = 9 + 4 + 6 = 19.
  3. n(U) = 19 + 3 = 22.
  4. n(A′) = 6 + 3 = 9 (everything not in A).

Answer: 13, 10, 19, 22 and 9.

Example 2: Cricket and football

Question: In a class of 40 students, 25 play cricket, 18 play football and 7 play both. How many play
(a) cricket only
(b) football only
(c) at least one of the sports
(d) neither?

  1. Put 7 in the overlap.
  2. Cricket only: 25 − 7 = 18. Football only: 18 − 7 = 11.
  3. At least one: 18 + 7 + 11 = 36 (or 25 + 18 − 7 = 36).
  4. Neither: 40 − 36 = 4. Check: 18 + 7 + 11 + 4 = 40. ✓

Answer:
(a) 18
(b) 11
(c) 36
(d) 4.

Venn diagram for cricket C and football F with 18 in C only, 7 in both, 11 in F only and 4 outside U CF 187114
Figure 2 – The Venn diagram for Example 2.

Example 3: Disjoint sets

Question: A = {even numbers from 1 to 10} and B = {odd numbers from 1 to 10}. Find n(A ∩ B) and n(A ∪ B).

  1. No number is both even and odd, so A and B are disjoint: n(A ∩ B) = 0.
  2. n(A ∪ B) = n(A) + n(B) = 5 + 5 = 10.

Answer: n(A ∩ B) = 0 and n(A ∪ B) = 10.

Example 4: Finding the overlap

Question: In a group of 30 students, 19 take Spanish, 15 take French and 2 take neither. How many take both?

  1. Students taking at least one language: 30 − 2 = 28.
  2. Use the formula: 28 = 19 + 15 − n(both), so n(both) = 34 − 28 = 6.
  3. Spanish only = 13, French only = 9. Check: 13 + 6 + 9 + 2 = 30. ✓

Answer: 6 students take both.

Example 5: Using x

Question: A survey of 35 people asked who likes doubles (D) and who likes roti (R). 20 like doubles, 16 like roti and 5 like neither. Let x be the number who like both. Find x.

  1. Doubles only = 20 − x and roti only = 16 − x.
  2. All the regions add to 35: (20 − x) + x + (16 − x) + 5 = 35.
  3. So 41 − x = 35, and x = 6.
  4. Doubles only = 14, roti only = 10. Check: 14 + 6 + 10 + 5 = 35. ✓

Answer: x = 6.

Common mistakes

  • Putting the whole of n(A) in the “A only” region. Always subtract the overlap first.
  • Adding n(A) and n(B) to get the union. This counts the overlap twice unless the sets are disjoint.
  • Forgetting the region outside both circles. All four regions must add up to n(U).
  • Mixing up ∩ and ∪. ∩ (like an “n” shape) is the overlap; ∪ (like a cup) holds everything in either set.
  • Reading “at least one” as “exactly one”. At least one includes those in both.

Practice questions

Try these yourself first, then open each answer to check your working.

1. P = {a, b, c, d, e} and Q = {a, e, i, o, u}. Find
(a) P ∩ Q
(b) P ∪ Q
(c) n(P ∪ Q).

Show answer

(a) {a, e}
(b) {a, b, c, d, e, i, o, u}
(c) 8.

2. Say whether each pair is a subset pair, disjoint or intersecting:
(a) A = {1, 2, 3}, B = {1, 2, 3, 4, 5}
(b) C = {2, 4, 6}, D = {1, 3, 5}
(c) E = {3, 6, 9}, F = {2, 3, 4}.

Show answer

(a) A ⊂ B (subset)
(b) Disjoint
(c) Intersecting (they share 3).

3. n(A) = 14, n(B) = 9 and n(A ∩ B) = 5. Find n(A ∪ B).

Show answer

14 + 9 − 5 = 18.

4. M and N are disjoint, n(M) = 12 and n(N) = 7. Find n(M ∪ N).

Show answer

19.

5. In a club of 50 members, 32 swim, 24 run and 10 do both. How many
(a) only swim
(b) only run
(c) do neither?

Show answer

(a) 22
(b) 14
(c) 50 − 46 = 4.

6. In a class of 36, 21 have a dog, 15 have a cat and 6 have neither. How many have both?

Show answer

36 − 6 = 30 have at least one, so both = 21 + 15 − 30 = 6.

7. n(U) = 40, n(A) = 22, n(B) = 25 and n((A ∪ B)′) = 3. Using x for n(A ∩ B), write an equation and find x.

Show answer

(22 − x) + x + (25 − x) + 3 = 40, so 50 − x = 40 and x = 10.

8. A Venn diagram shows A only = 2x, A ∩ B = 5, B only = x + 1, outside = 4. If n(U) = 31, find x and n(A).

Show answer

2x + 5 + x + 1 + 4 = 31, so 3x + 10 = 31 and x = 7. n(A) = 14 + 5 = 19.

Quick check

1. Write the formula for n(A ∪ B).

Show answer

n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

2. What is n(A ∩ B) if A and B are disjoint?

Show answer

0.

3. If n(A) = 10 and n(A ∩ B) = 4, how many are in A only?

Show answer

6.

Previous: Solving Inequalities and Showing Answers on a Number Line and in Set Builder Notation

Next: Relations and Arrow Diagrams: Domain, Range, Mappings and Functions

Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 2, Term 1, Strand 2.2 Sets, Relations and Functions, Topic 2.2.1 Sets – Learning Outcomes 2.2.1.1 (subsets, disjoint and intersecting sets), 2.2.1.2 (counting the union and intersection), 2.2.1.3 (representing sets on Venn diagrams), 2.2.1.4 (the union of intersecting and disjoint sets), 2.2.1.5 (two-set problems given the number of elements) and 2.2.1.6 (two-set word problems).

Updates

2026/10/23

– First version of this lesson.

2026/10/25

– Put each lettered part of the questions and answers on its own line.

© 2026 Vedesh Kungebeharry. All rights reserved.

Sets for Beginners: Set Notation, Types of Sets, Subsets and Venn Diagrams

Aven BrisailLeave a comment

A set is a collection of objects, numbers or ideas that belong together. Each object in a set is called a member or an element. For example, the colours of the flag of Trinidad and Tobago form the set {red, white, black}. Sets help us sort and classify things, and they are the starting point for Venn diagrams, probability and logic. This lesson covers naming and describing sets, the main types of sets, subsets, and the ideas of union, intersection and complement shown on Venn diagrams.

Naming and describing a set

  • Name a set with a capital letter and put its members inside curly brackets { }, separated by commas: V = {a, e, i, o, u}.
  • A set must be well defined: you must be able to say for certain whether something belongs. “Even numbers” is well defined; “nice songs” is not.
  • You can define a set in two ways: by listing the members, V = {a, e, i, o, u}, or by describing them in words, V = {vowels in the English alphabet}.
  • The symbol ∈ means “is a member of” and ∉ means “is not a member of”: e ∈ V, but b ∉ V.
  • n(A) means the number of members in set A. For the vowels, n(V) = 5.
  • Each member is listed only once, and the order does not matter: {1, 2, 3} is the same set as {3, 1, 2}.

Types of sets

TypeMeaningExample
Empty (null) setHas no members. Written { } or ∅.{months with 32 days} = { }
Finite setYou could count all its members.{pupils in your class}
Infinite setThe members go on forever. Use “…” in a list.{multiples of 5} = {5, 10, 15, …}
Equal setsExactly the same members (A = B).{letters in LISTEN} = {letters in SILENT}
Equivalent setsThe same number of members, n(A) = n(B), even if the members differ.{1, 2, 3} and {a, b, c}

Subsets

If every member of set A is also in set B, then A is a subset of B, written A ⊂ B (some books write A ⊆ B). For example, {2, 4} ⊂ {1, 2, 3, 4, 5}. Two useful facts: the empty set is a subset of every set, and every set is a subset of itself. When we classify the members of a set, we divide it into subsets: {1, 2, 3, 4, 5, 6} can be split into the subsets {odd numbers} = {1, 3, 5} and {even numbers} = {2, 4, 6}.

Universal set, complement, union and intersection

  • The universal set, U, contains every member being talked about in a question – for example U = {whole numbers from 1 to 12}.
  • The complement of A, written A′, is everything in U that is not in A.
  • The union A ∪ B is everything in A or B or both (join the sets together, without repeats).
  • The intersection A ∩ B is everything in both A and B (the overlap).
  • Sets that share at least one member are intersecting sets. Sets with no members in common are disjoint: A ∩ B = { }.

Venn diagrams

A Venn diagram shows sets as circles inside a rectangle that stands for U. Overlapping circles show intersecting sets; separate circles show disjoint sets; a circle drawn completely inside another shows a subset. Members outside every circle are still in U – do not forget to write them in the rectangle.

Stained-glass window showing three overlapping coloured circles of a Venn diagram
Figure 1 – A Venn diagram in a stained-glass window at Gonville and Caius College, Cambridge, honouring John Venn, who made these diagrams famous in 1880. Image: Schutz (photo); stained glass designed by Maria McClafferty, CC BY-SA 2.5, via Wikimedia Commons.

TRY ITVenn diagram builder

Choose a rule for set A and for set B. Every number from 1 to 20 drops into the correct region, and the sets are written out for you in set notation.

Universal set: U = {whole numbers from 1 to 20}

Venn diagram of sets A and B U AB

Worked examples

Example 1: Listing a set

Question: List the members of A = {odd numbers between 2 and 12} and find n(A).

  1. “Between 2 and 12” means we do not include 2 or 12.
  2. The odd numbers in that range are 3, 5, 7, 9, 11.

Answer: A = {3, 5, 7, 9, 11}, n(A) = 5.

Example 2: Describing a set in words

Question: Describe the set {2, 4, 6, 8} in words.

  1. Look for what the members have in common: they are all even, start at 2 and stop at 8.

Answer: {even numbers from 2 to 8} (or {the first four even numbers}).

Example 3: Union, intersection and complement

Question: U = {whole numbers from 1 to 12}, A = {factors of 12} and B = {even numbers}. List A, B, A ∩ B, A ∪ B, A′ and (A ∪ B)′, and draw the Venn diagram.

  1. A = {1, 2, 3, 4, 6, 12} and B = {2, 4, 6, 8, 10, 12}.
  2. A ∩ B (in both) = {2, 4, 6, 12}. These go in the overlap.
  3. A only = {1, 3}; B only = {8, 10}.
  4. A ∪ B = {1, 2, 3, 4, 6, 8, 10, 12}.
  5. A′ (in U but not in A) = {5, 7, 8, 9, 10, 11}.
  6. (A ∪ B)′ (outside both circles) = {5, 7, 9, 11}.
  7. Check: n(A) + n(B) − n(A ∩ B) = 6 + 6 − 4 = 8 = n(A ∪ B) ✔

Answer: Venn diagram: 1 and 3 in A only; 2, 4, 6, 12 in the overlap; 8 and 10 in B only; 5, 7, 9, 11 outside the circles. (Choose “Factors of 20”, “Even numbers” and other rules in the builder above to practise more of these.)

Example 4: Equal or equivalent?

Question: Are {letters in LISTEN} and {letters in SILENT} equal sets? Are {1, 2, 3} and {a, b, c} equal?

  1. LISTEN gives {L, I, S, T, E, N} and SILENT gives {S, I, L, E, N, T} – the same six letters, so the sets are equal.
  2. {1, 2, 3} and {a, b, c} have different members, but both have 3 members.

Answer: The first pair are equal sets. The second pair are equivalent but not equal.

Example 5: Listing all subsets

Question: List every subset of {p, q, r}.

  1. No members: { }.
  2. One member: {p}, {q}, {r}.
  3. Two members: {p, q}, {p, r}, {q, r}.
  4. Three members: {p, q, r}.

Answer: 8 subsets. (A set with n members has 2n subsets: 23 = 8.)

Common mistakes

  • Repeating members. The letters in “BANANA” form the set {B, A, N}, not {B, A, N, A, N, A}.
  • Mixing up ∪ and ∩. Think “∪ for union = everything joined”; “∩ looks like an n for intersection = only the overlap”.
  • Forgetting members outside the circles. Every member of U must appear somewhere in the rectangle.
  • Confusing ∅ with {0}. ∅ has no members; {0} has one member, the number 0.
  • Using ∈ for sets. 3 ∈ {1, 3, 5} (a member), but {3} ⊂ {1, 3, 5} (a subset).

Practice questions

Try these yourself first, then open each answer to check your working.

1. List the set C = {prime numbers less than 20}. What is n(C)?

Show answer

C = {2, 3, 5, 7, 11, 13, 17, 19}, n(C) = 8.

2. Which of these sets are infinite?
(a) {multiples of 5}
(b) {pupils in your school}
(c) {even numbers less than 1,000,000}

Show answer

Only (a). Sets (b) and (c) are very large but could be counted, so they are finite.

3. U = {1, 2, 3, …, 10}, P = {1, 2, 3, 4, 5} and Q = {4, 5, 6, 7}. List P ∩ Q, P ∪ Q and Q′.

Show answer

P ∩ Q = {4, 5}; P ∪ Q = {1, 2, 3, 4, 5, 6, 7}; Q′ = {1, 2, 3, 8, 9, 10}.

4. Are {even numbers} and {odd numbers} disjoint or intersecting? Explain.

Show answer

Disjoint – no whole number is both even and odd, so their intersection is empty.

5. How many subsets does {a, b, c, d} have?

Show answer

24 = 16 subsets (including { } and the set itself).

6. True or false?
(a) 9 ∈ {square numbers}
(b) {3} ⊂ {1, 3, 5}
(c) 4 ∈ {odd numbers}

Show answer

(a) True
(b) True
(c) False.

7. In a class of 30 pupils, 18 play football, 15 play cricket and 8 play both. Draw a Venn diagram. How many play neither?

Show answer

Football only: 18 − 8 = 10. Cricket only: 15 − 8 = 7. Both: 8. That is 10 + 8 + 7 = 25 pupils, so 30 − 25 = 5 play neither.

Quick check

1. What does n(A) mean?

Show answer

The number of members (elements) in set A.

2. What is the difference between equal and equivalent sets?

Show answer

Equal sets have exactly the same members; equivalent sets just have the same number of members.

3. Which symbol means “intersection”?

Show answer

∩ (the members in both sets).

4. Where do you write members of U that are in neither A nor B?

Show answer

Inside the rectangle but outside both circles.

Next: Tally Charts, Frequency Tables, Pictographs and Block Graphs

Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 1, Term 1, Strand 1.2 Sets, Relations and Functions, Topic 1.2.1 Sets – Learning Outcomes 1.2.1.1 (classifying, describing and naming sets), 1.2.1.2 (defining sets by listing or describing), 1.2.1.3 (dividing a set into subsets), 1.2.1.4 (empty, equal, equivalent, finite and infinite sets), 1.2.1.5 (universal sets, complements, union, intersecting sets, subsets and disjoint sets) and 1.2.1.6 (Venn diagrams for two sets).

Attributions to media used in this post:

Venn-stainedglass-gonville-caius.jpg – Schutz (photo); stained glass designed by Maria McClafferty – CC BY-SA 2.5 (via Wikimedia Commons)

Updates

2026/10/25

– Put each lettered part of the questions and answers on its own line.

© 2026 Vedesh Kungebeharry. All rights reserved.