Mathematics • Year 9 • Unit 4 • Lesson 15

Venn Diagrams and Set Notation

Build fluency with set notation (∪, ∩, ′), Venn diagrams and the addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B) — from a worked example through guided practice to eight independent problems.

Build · I Do / We Do / You Do

1. I do — fully worked example

Read every step. The reason on the right tells you why, not just what.

Problem. In a group of 50 people, 30 like tea, 25 like coffee and 15 like both. Draw a Venn diagram and find how many like neither. Then find P(tea or coffee).

Step 1 — Fill in the intersection first.

Both tea AND coffee = 15. Put 15 in the overlap region.

Reason: the overlap (A ∩ B) is always filled first so you don't accidentally double-count.

Step 2 — Tea only and Coffee only.

Tea only = 30 − 15 = 15. Coffee only = 25 − 15 = 10.

Reason: 30 includes the 15 in the overlap, so subtract to get the tea-only region.

Step 3 — Sketch the Venn diagram.

   +-------------------------- 50 ----------------+
   |    .--- Tea ---.   .--- Coffee ---.          |
   |   /   15      /\ \  \   10        \          |
   |  |          /   15  |                        |
   |   \         \_____/ /                        |
   |    `-------´       `-------´                 |
   |                                       Neither: 10
   +-----------------------------------------------+

Step 4 — Neither = total − everything inside the circles.

Inside = 15 + 15 + 10 = 40. Neither = 50 − 40 = 10.

Step 5 — Apply the addition rule.

P(tea) = 30/50 = 0.6, P(coffee) = 25/50 = 0.5, P(tea ∩ coffee) = 15/50 = 0.3.

P(tea ∪ coffee) = 0.6 + 0.5 − 0.3 = 0.8.

Reason: subtract the intersection so the overlap is not counted twice.

Answer: 10 people like neither. P(tea or coffee) = 0.8 (= 40/50).

Stuck? Revisit lesson § "Venn Diagrams" — always fill the overlap first, then subtract.

2. We do — fill in the missing steps

Same structure as Section 1, but with the working faded. Fill in each blank line. 5 marks

Problem. In a class of 30, 18 play tennis, 15 play basketball, 8 play both. Find tennis only, basketball only, neither, and P(tennis ∪ basketball).

Step 1 — Overlap: tennis ∩ basketball = ______ .

Step 2 — Tennis only: 18 − ______ = ______ .

Step 3 — Basketball only: 15 − ______ = ______ .

Step 4 — Sum inside circles:

Inside = ______ + ______ + ______ = ______ .

Step 5 — Neither:

Neither = 30 − ______ = ______ .

Step 6 — Addition rule:

P(T) = ______ ÷ 30 = ______ . P(B) = ______ ÷ 30 = ______ . P(T ∩ B) = ______ ÷ 30 = ______ .

P(T ∪ B) = P(T) + P(B) − P(T ∩ B) = ______ + ______ − ______ = ______ .

Stuck? Revisit lesson § "Think First" — 18 + 15 − 8 = 25 play at least one, so 30 − 25 = 5 play neither.

3. You do — independent practice

Show working under each problem. Foundation = read notation; Standard = small Venn; Extension = addition rule and missing values.

Foundation — read the notation

3.1 A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. List A ∪ B. 1 mark

3.2 Same sets as 3.1. List A ∩ B. 1 mark

3.3 If the universal set is U = {1, 2, 3, 4, 5, 6, 7, 8} and A = {2, 4, 6, 8}, list A′ (the complement). 1 mark

3.4 If P(A) = 0.4, P(B) = 0.3 and A and B are mutually exclusive, find P(A ∪ B). 1 mark

Standard — small Venn diagrams

3.5 In a class of 40, 25 play sport, 20 play music, 12 do both. How many do neither? 2 marks

3.6 P(A) = 0.3, P(B) = 0.4, P(A ∩ B) = 0.1. Find P(A ∪ B) and P(A only) (i.e. A but not B). 3 marks

Extension — addition rule / missing values

3.7 A single card is drawn from a 52-card deck. Find P(heart ∪ king) using the addition rule. 3 marks

3.8 Given P(A ∪ B) = 0.8, P(A) = 0.5, P(B) = 0.4, find P(A ∩ B). 2 marks

Stuck on 3.8? Rearrange the addition rule: P(A ∩ B) = P(A) + P(B) − P(A ∪ B).

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Section 2 — We do (30 students, 18 tennis, 15 basketball, 8 both)

Overlap = 8. Tennis only = 18 − 8 = 10. Basketball only = 15 − 8 = 7.
Inside circles = 10 + 8 + 7 = 25. Neither = 30 − 25 = 5.
P(T) = 18/30 = 0.6. P(B) = 15/30 = 0.5. P(T ∩ B) = 8/30 ≈ 0.267.
P(T ∪ B) = 0.6 + 0.5 − 0.267 ≈ 0.833 (or exactly 25/30 = 5/6).

3.1 — A ∪ B

A ∪ B = {1, 2, 3, 4, 5, 6}.

3.2 — A ∩ B

A ∩ B = {3, 4}.

3.3 — A′

A′ = everything in U not in A = {1, 3, 5, 7}.

3.4 — Mutually exclusive A ∪ B

For mutually exclusive events P(A ∩ B) = 0, so P(A ∪ B) = 0.4 + 0.3 = 0.7.

3.5 — 40 students, sport ∩ music

Sport only = 25 − 12 = 13. Music only = 20 − 12 = 8. Inside = 13 + 12 + 8 = 33. Neither = 40 − 33 = 7.

3.6 — Addition rule

P(A ∪ B) = 0.3 + 0.4 − 0.1 = 0.6. P(A only) = P(A) − P(A ∩ B) = 0.3 − 0.1 = 0.2.

3.7 — P(heart ∪ king)

P(heart) = 13/52, P(king) = 4/52, P(heart ∩ king) = 1/52 (the king of hearts).
P(heart ∪ king) = 13/52 + 4/52 − 1/52 = 16/52 = 4/13.

3.8 — Find P(A ∩ B)

P(A ∩ B) = P(A) + P(B) − P(A ∪ B) = 0.5 + 0.4 − 0.8 = 0.1.