Venn Diagrams with Two Events

Lesson 3 of 4: Sets

In this lesson:

• Draw a correctly labelled two-circle Venn diagram
• Place elements in the right regions using the intersection-first rule
• Read set values from a completed diagram
• Solve word problems with Venn diagrams

What You Will Be Able To Do

1. Draw a two-circle Venn diagram with a labelled universal set
2. Place elements correctly using the intersection-first rule
3. Read , , , , and from a diagram
4. Solve real-life word problems using a Venn diagram

Where Does Each Learner Stand?

In your class, some learners play football, some play volleyball, some play both, and some play neither.

• Aisha: football only
• Brian: both football and volleyball
• Christine: volleyball only
• David: neither

Which group does each person belong to? Can you see four distinct groups?

Building the Diagram: Step 1

Draw a rectangle and label it

= the universal set — all the learners in P7

Every learner in the class belongs inside this rectangle — no exceptions.

Building the Diagram: Steps 2 and 3

• Draw circle A inside E: = {learners who like football}
• Draw circle B overlapping A: = {learners who like volleyball}
• The overlap is where both circles meet

The Four Regions of the Venn Diagram

Every element of belongs in exactly one region.

Reading the Regions from a Description

Which region does each description fit?

1. A learner who likes football but not volleyball
2. A learner who likes both sports
3. A learner who likes volleyball but not football
4. A learner who likes neither sport

Name the region: A only, , B only, or neither.

Quick Check — Which Region?

= {like mangoes}, = {like bananas}, = {P7 learners}

Where does each learner go?

1. Esther likes mangoes only
2. David likes both
3. Christine likes bananas only
4. Aisha likes neither

Answers: 1 — A only, 2 — A∩B, 3 — B only, 4 — neither

The Placement Problem: Counting Twice

, ,

The number 6 is even AND a multiple of 3.

If we place 6 in A first and also try to place it in B, we count it twice.

How do we avoid this?

The Intersection-First Rule for Placement

Fill in this order:

1. — in both — place first
2. A only — in A, not
3. B only — in B, not
4. Neither — in E, not yet placed

Check: every element placed once, none twice.

Worked Example: Step by Step

,

Step 1 — →

Step 2 — A only →

Step 3 — B only →

Step 4 — neither →

Completed Diagram and Verification Check

Verification check:

Region counts: , , ,

Reading Cardinalities from the Diagram

From the completed diagram:

Quick Check — Read the Diagram

Same diagram (, = even, = mult. of 3):

1. ?
2. ?
3. How many are in neither?
4. Verify ?

Answers: 1 → 7, 2 → 7, 3 → 3, 4 → 10 ✓

Word Problems: Only Totals Given

In a class of 30 learners, 18 study English, 15 study Luganda, and 7 study both.

Strategy: intersection-first, using algebra

• , , ,
• Place 7 in the intersection first, then subtract for exclusive regions

Word Problem: Steps 1 and 2

Let = English, = Luganda,

Step 1 — Intersection:

Step 2 — Exclusive:

Word Problem: Steps 3 and 4

Step 3 — neither:

Step 4 — Verify: ✓

The Complement of a Set:

The complement of A, written , is:

All elements of that are not in

On the diagram, is everything outside circle A but inside E:

Check: always.

Reading from the Diagram

From the English/Luganda diagram:

• (do not study English)
• ✓
• ; ✓

A set and its complement cover all of .

Practice: A Second Word Problem to Try

25 learners, 14 play netball, 11 play tennis, 5 play both.

Draw the Venn diagram and find:

1. Netball only?
2. Tennis only?
3. Neither?
4. Verify: four regions sum to 25.

Answers: netball = 9, tennis = 6, neither = 5; 9+5+6+5 = 25 ✓

Watch Out — Common Errors

Error 1: Double-counting the intersection

• ✓ Intersection elements go in the centre only

Error 2: Forgetting "neither"

• ✓ Four regions must sum to

Error 3: Confusing with

• ✓ = three regions; = one

Key Takeaways from This Lesson

✓ Four regions: A only, , B only, neither
✓ Intersection-first rule: place before the exclusive regions
✓ Verify: all four region counts must sum to
✓ Reading: A only + centre; three inner; outside A