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📐 Mathematics - Class 7
Lesson 11 · Sets

Operations on Sets

MathematicsSubject
14 minEstimated read
Two sets, and two questions

Once you have two sets side by side, there are only two natural questions. Is it in this one OR that one? Is it in this one AND that one? The first builds the union, the second builds the intersection.

The four operations

Name Symbol The question it asks
UnionA ∪ BIs it in A OR in B, or in both?
IntersectionA ∩ BIs it in A AND in B?
DifferenceA − BIs it in A but NOT in B?
ComplementA′Everything in the universal set that is not in A
A way to remember the two symbols

The union symbol is shaped like a cup, and a cup holds everything you pour into it. The intersection symbol is the same shape upside down, like a cap, and a cap only covers the part that both share.

The universal set

The universal set, written U, is the collection of all the objects under discussion. If U is the numbers 1 to 10 and A is the even ones, then A complement is {1, 3, 5, 7, 9}. The same set A can have completely different complements depending on what U is.

Always look for U first

A question that asks for a complement must tell you the universal set somewhere. Read the whole question, underline U, and write it at the top of your working.

Counting a union without counting twice

If 18 play football, 12 play volleyball and 5 play both, the answer is not 30. The 5 who play both have been counted twice. Subtract the overlap once and you get 25.

\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]

Why the subtraction is there

Every element of the intersection sits inside A and inside B, so adding the two totals counts it twice. Taking the intersection away once fixes exactly those elements and leaves the rest alone. It is a repair, not a formula to memorise.

Order matters for one of them

Operation Does the order matter?
A ∪ BNo. A ∪ B = B ∪ A
A ∩ BNo. A ∩ B = B ∩ A
A − BYES, and this is a common trap
A minus B is not B minus A

If A = {1, 2, 3, 4} and B = {3, 4, 5}, then A − B = {1, 2} but B − A = {5}. Read A − B as what is left of A when B is taken out, and the direction becomes obvious.

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Worked Example 1

U = {1, 2, ..., 10}, A = {2, 4, 6, 8, 10}, B = {1, 2, 3, 4, 5}. Find A ∪ B, A ∩ B, A − B and A′.

Union: collect everythingA ∪ B = {1, 2, 3, 4, 5, 6, 8, 10}
Intersection: keep only the shared onesA ∩ B = {2, 4}
Difference: A with B taken outA − B = {6, 8, 10}
Complement: everything in U outside AA′ = {1, 3, 5, 7, 9}
Worked Example 2

In a class of 40, 25 study science, 18 study computing, and 8 study both. How many study neither?

Step 1: find how many study at least onen(A ∪ B) = 25 + 18 - 8 = 35
Step 2: take that from the whole classNeither = 40 - 35 = 5 students
The word neither is a complementNeither means outside the union, so it is the complement of A ∪ B.
Sense check17 + 10 + 8 + 5 = 40, the whole class. Everything is accounted for exactly once.
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Activity 11.2

Survey the class and find out how many were double counted

What you needNothing but the class itself. Choose two activities that a fair number of people do and that genuinely overlap.
MethodCount the hands for the first activity, then the second, then for BOTH, then count the whole class.
Then predict before you checkPredict how many do at least one, and how many do neither. Write both down. Only then count the neither group.
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1

Union A ∪ B: everything in A OR in B, or in both. The cup collects everything.

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1Define union and intersection, and give an example of each using two sets of your own.
2Why is n(A ∪ B) not simply n(A) + n(B)?
3Explain why A − B and B − A are usually different sets, with an example.
4What is the universal set, and why does a complement depend on it?
5In a class of 40, 25 study science, 18 study computing and 8 study both. Work out how many study only science, only computing, and neither.

Question 1 of 10

1A ∪ B contains everything that is
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Slide 1 of 4
Unit 11 · Set Operations

Set Operations

Mathematics · Grade 7

What You Will Be Able To Do

∪

Work out unions, intersections, differences and complements

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Recognise which operation a word problem is really asking for

➖

Count a union without counting the overlap twice

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Solve survey problems: how many do only one, both, or neither

“

Once you have two sets side by side, there are only two natural questions. Is it in this one OR that one? Is it in this one AND that one?

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Presenter notes: Open with a live count. Ask everyone who plays football to stand, count them and write the number up. Sit them down, ask everyone who plays volleyball to stand, count and write that up. Add the two numbers out loud and ask whether that is how many people play at least one sport. Somebody will object that a few people stood up twice. That objection is the entire lesson, and it has arrived before you have written a single symbol.