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).

Both = 15, so tea-only = 30−15 = 15 and coffee-only = 25−15 = 10; neither = 50−40 = 10.

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.