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

• 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).

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.

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.

Quick check

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.