Grade 10 Sets: Short Notes

10 ශ්‍රේණිය · 17 පරිච්ඡේදය · කුලක

96%

of Grade 10 term tests include it (52 of 54)

711

marks in past papers (120 questions)

11 min

full lesson with quick checks

🎯 What you'll learn

Write a set by description, by listing, in generating form and in a Venn diagram
Use ξ, ∈, ⊂, A', A ∪ B and A ∩ B correctly
Shade and name any region of a two-set Venn diagram, and describe it in words
Use n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Solve two-set word problems with a Venn diagram

Cricket or football?

In a Grade 10 class of 40 in Matara, 25 students follow cricket, 18 follow football and 8 follow both. How many follow neither? 25 + 18 is already more than 40, so simply adding can't be right. Sets and one neat diagram sort it out in seconds.

Four ways to write a set

Take the even numbers between 1 and 11.• Description: A is the set of even numbers between 1 and 11.• Listing elements: A = {2, 4, 6, 8, 10}• Generating form: A = {x : x is even, 1 < x < 11}• Venn diagram: the elements written inside a closed curve.n(A) is the number of elements: n(A) = 5.

Reading generating form

{x : x is a whole number, 3 ≤ x < 8} reads: "the set of all x such that x is a whole number from 3 up to, but not including, 8."So it is {3, 4, 5, 6, 7}.Watch the signs: ≤ includes the end number, < does not.

The symbols you need

• ξ universal set: everything we are talking about• ∈ 'is an element of': 4 ∈ A• B ⊂ A: B is a subset of A (every element of B is in A)• A' complement: elements of ξ not in A• A ∪ B union: in A or B (or both)• A ∩ B intersection: in both A and B

The four regions

Two overlapping circles split ξ into four parts: A only (A ∩ B'), both (A ∩ B), B only (A' ∩ B), and outside both, (A ∪ B)'.

Say it in words

Exams ask you to describe regions. Let ξ = students in a class, G = girls, D = students who dance.• G ∩ D: girls who dance• G ∩ D': girls who don't dance• G' ∩ D: boys who dance (G' = not girls = boys)• (G ∪ D)': boys who don't danceRead ' as not and ∩ as and.

⚠️ Shading complements

In one O/L paper only about 1 in 10 candidates shaded A' ∪ B' correctly. Build it in steps: shade A' (everything outside A), then B' (everything outside B). The union is every part shaded at least once: everything except the overlap. So A' ∪ B' = (A ∩ B)'.Check each region one by one: is it in A? Is it in B?

One set inside another

If every element of B is in A, draw B inside A: B ⊂ A. Then A ∩ B = B and A ∪ B = A. The amber ring is A ∩ B' (in A, not in B).

🧮 Counting a union

Adding n(A) and n(B) counts the overlap twice, so subtract it once. Use it when you know three of these numbers and need the fourth.

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

✏️ Back to the cricket class

n(ξ) = 40, cricket n(C) = 25, football n(F) = 18, both n(C ∩ F) = 8. How many follow neither sport?

1n(C ∪ F) = 25 + 18 − 8 = 35 follow at least one sport.
2Neither = n(ξ) − n(C ∪ F) = 40 − 35 = 5.
3In set notation: n((C ∪ F)') = 5.

✅ 5 students

⚠️ '25 follow cricket' includes 'both'

A classic error is writing 25 in the 'cricket only' part. The 25 includes the 8 who follow both. Fill the overlap first, then subtract: cricket only = 25 − 8 = 17, football only = 18 − 8 = 10. Finally check that the four regions add up to n(ξ): 17 + 8 + 10 + 5 = 40 ✓

✏️ A full Venn word problem

Of 60 Grade 10 students in a Kandy school, 32 are in the Science Society (S), 25 in the Sports Club (C) and 12 in both. Find (i) the number in neither, (ii) the number in exactly one, (iii) n(S').

1Overlap first: n(S ∩ C) = 12.
2S only = 32 − 12 = 20; C only = 25 − 12 = 13.
3(i) Neither = 60 − (20 + 12 + 13) = 15.
4(ii) Exactly one = 20 + 13 = 33.
5(iii) S' is everything outside S: 13 + 15 = 28.

✅ (i) 15 (ii) 33 (iii) 28

📌 Remember

• A set can be described, listed, written in generating form {x : ...} or drawn in a Venn diagram.• ' means not, ∩ means and (both), ∪ means or (at least one).• Fill the overlap first, then the 'only' parts, then 'neither'.• n(A ∪ B) = n(A) + n(B) − n(A ∩ B).• Check: the four regions add up to n(ξ).

🧾 Quick revision sheet

What is ξ?	The universal set: all the elements being considered.
A ∩ B	Intersection: elements in both A and B.
A ∪ B	Union: elements in A or B or both.
A'	Complement: elements of ξ that are not in A.
Formula for n(A ∪ B)	n(A) + n(B) − n(A ∩ B)
A ∩ B' in words	In A but not in B (A only).
If B ⊂ A, what are A ∩ B and A ∪ B?	A ∩ B = B and A ∪ B = A
List {x : x is odd, 1 ≤ x ≤ 9}	{1, 3, 5, 7, 9}
A' ∪ B' is the same as…	(A ∩ B)': everything except the overlap.