Grade 11 Sets: Short Notes

11 ශ්‍රේණිය · 24 පරිච්ඡේදය · කුලක

57%

of Grade 11 term tests include it (35 of 61)

350

marks in past papers (35 questions)

11 min

full lesson with quick checks

🎯 What you'll learn

Name the eight regions of a three-set Venn diagram
Describe a region in words and in set notation, and shade it
Fill a three-set Venn diagram from data, centre first
Draw subsets correctly inside a Venn diagram
Solve problems with unknown regions and find probabilities

Rice, dhal or sugar?

A grocery in Matara checks what 60 customers bought on a Saturday: rice, dhal, sugar, or some mix of the three. With three items, two circles are not enough. Grade 11 Venn diagrams use three circles and eight regions, and they turn up in Paper II almost every year.

Eight regions

p = A only, q = B only, r = C only; s = A and B only, t = B and C only, u = A and C only; v = all three (A ∩ B ∩ C); w = none, (A ∪ B ∪ C)'.

Naming regions

Read ∩ as 'and', ∪ as 'or', ' as 'not'.• A ∩ B ∩ C' = in A and B but not C (region s)• A ∩ (B ∪ C)' = in A but in neither B nor C (region p)• (A ∩ B) ∪ (A ∩ C) = in A and at least one other (s, u, v)• 'Exactly two' = s + t + u (not v!)

Fill from the centre out

1. Write the all three number in the centre.2. For each pair, 'both A and B' includes the centre, so A and B only = n(A ∩ B) − centre.3. Each 'only' region = n(A) − the three other parts of A.4. None = n(ξ) − everything inside the circles.Finally check that all 8 regions add up to n(ξ).

✏️ The Matara grocery

Of 60 customers, 30 bought rice (R), 25 dhal (D), 28 sugar (S). 10 bought rice and dhal, 8 dhal and sugar, 12 rice and sugar, and 5 bought all three. Complete the Venn diagram and find how many bought none.

1Centre: R ∩ D ∩ S = 5.
2Pairs only: R and D 10 − 5 = 5; D and S 8 − 5 = 3; R and S 12 − 5 = 7.
3Only: R = 30 − (5 + 5 + 7) = 13; D = 25 − 13 = 12; S = 28 − 15 = 13.
4Inside circles: 13 + 12 + 13 + 5 + 3 + 7 + 5 = 58.
5None = 60 − 58 = 2.

✅ 2 customers bought none

⚠️ 'Rice and dhal' includes the centre

Writing 10 straight into the rice-and-dhal region is the most common error. The 10 includes the 5 who bought all three, so that region gets 10 − 5 = 5. Examiners say many students place numbers in the wrong regions and every later part goes wrong. Always fill the centre first.

When one set is inside another

ξ = students, B = boys, P = prefects, F = footballers. If all prefects are boys, P ⊂ B, so draw P completely inside B. F can still cross B. Spot words like 'all' or 'every' before you draw.

⚠️ Watch the word 'not'

Examiners report that students misread 'not' and shade or count the wrong region. Break the words into pieces: 'girls who do not play netball but sing' = G ∩ N' ∩ S. Shade G, then remove N, then keep only the part inside S. Tick each of the 8 regions: yes or no?

✏️ Regions with an unknown

In a class of 45 in Badulla, students do Dancing (D), Music (M) and Art (A). Twice as many do dancing only as music only. Let music only = x. The other regions: art only 5, D and M only 4, M and A only 3, D and A only 2, all three 1, none 3. Find x, then the probability that a random student does exactly one activity.

1Dancing only = 2x.
2Add all regions: 2x + x + 5 + 4 + 3 + 2 + 1 + 3 = 3x + 18.
33x + 18 = 45, so 3x = 27 and x = 9.
4Exactly one: 18 + 9 + 5 = 32.
5Probability = 32/45.

✅ x = 9, probability 32/45

📌 Remember

• Three sets give 8 regions, including 'none' outside.• Fill the centre first, then the 'two only' regions, then 'only', then 'none'.• ∩ = and, ∪ = or, ' = not. Split long descriptions into pieces.• 'All X are Y' means X ⊂ Y: draw X inside Y.• Unknowns: add all regions, set equal to n(ξ), solve.

🧾 Quick revision sheet

How many regions in a 3-set Venn diagram?	8 (7 inside the circles + 1 outside)
A ∩ B ∩ C' in words	In A and B but not in C.
A ∩ (B ∪ C)' in words	A only: in A but in neither B nor C.
(A ∪ B ∪ C)'	The region outside all three circles: none of them.
'Exactly two' regions	The three 'two only' regions. Not the centre.
Which region do you fill first?	The centre: in all three sets.
'All prefects are boys' in symbols	P ⊂ B: draw P inside B.