Page 1

Teacher Abi Study Notebook

I DON’T UNDERSTAND...Sets, Subsets, Set Operations & Venn Diagrams

Looks like thisA = {2, 4, 6, 8}6 ∈ A

And thisTwo sets overlapA ∩ B

Okay. What are all these braces, circles, and symbols actually saying?

That is what this notebook is for. We’ll build the idea first, attach the symbol second, and use plenty of examples until the picture and the notation say the same thing.

Teacher Abi’s Note

Don’t memorize ∪ = union and ∩ = intersection and hope for the best. Learn what part of the set the question is asking you to look at.

See it firstConcrete examples before formal notation.

Try it right awayExample → your turn → feedback.

Use both directionsSets → diagram and diagram → operation.

Still completely lost?You don’t have to figure it out alone. Book a review session with Teacher Abi.

Book a Review Session

START HERE

What do you already know?

Don’t study anything yet. Just try these. It’s completely fine if you don’t know one.

1. A = {2,4,6,8}. Is 6 an element of A?

2. A = {1,2,3}, B = {3,4}. What is A ∩ B?

3. A set has 3 elements. How many subsets does it have?

Set Card 1

What even is a set?

Compare these two collections.

A. Months with 30 days

B. The nicest songs ever made

Which collection has a rule that everyone can check?

Now give the idea its math name

A set is a well-defined collection of objects called elements. Well-defined means we can decide whether something belongs.

Example

A = {2, 4, 6, 8}

The elements are 2, 4, 6, and 8. The braces { } hold the elements.

Set Card 2

Is it IN or NOT IN?

A = {2, 4, 6, 8}

Meaning first

SEE

6 appears inside A.

WRITE

6 ∈ A

READ

“6 is an element of A.”

SEE

5 does not appear inside A.

WRITE

5 ∉ A

B = {3,6,9,12}. Is 9 an element of B?

Using the same B, complete: 10 ___ B

Tap a symbol — no special keyboard needed.

Set Card 3

The same set can be written two ways.

Roster formList the elements.

{0,1,2,3,4}

Set-builder formDescribe the rule.

{x | x is a whole number, x < 5}

Worked example

Whole numbers less than 5

Set-builder: {x | x is a whole number, x < 5}

Start at 0 because these are whole numbers.

Stop before 5.

Answer

{0,1,2,3,4}

C = {2,4,6,8}. Which description matches?

Set Card 4

One object or a little set?

A = {1, 2, 3, 4}

Element

2 ∈ A
2 is one object inside A.

Subset

{2} ⊆ A
{2} is itself a set, and its element is in A.

Watch the braces. 2 and {2} are not the same thing.

Which statement describes {1,3} correctly?

Subset Card 1

A subset is a smaller set made from another set.

A = {a,b,c}

The rule

A set B is a subset of A when every element of B is also in A. This notebook uses ⊆ for “is a subset of” and ⊂ only for a proper subset.

{a,c}

Both a and c are in A.

{a,c} ⊆ A

{a,d}

d is not in A.

{a,d} ⊄ A

Which one is a subset of A?

Subset Card 2

What are the subsets? Let’s actually build them.

Start with two elements

B = {a,b}

A subset can choose neither, one, or both elements.

Choose neither: ∅

Choose one: {a}, {b}

Choose both: {a,b}

ALL

∅, {a}, {b}, {a,b}

Two facts to remember:
1. The empty set ∅ is a subset of every set.
2. Every set is a subset of itself.
That is why both ∅ and the whole set belong in the complete list.

Which list contains all subsets of {x,y}?

Subset Card 3

Now you build the list.

C = {1,2}

Use the same pattern

Neither → one at a time → both.

Before checking, write all the subsets of C on paper.

∅, {1}, {2}, {1,2}
There are 4 subsets.

How many subsets did you list?

Subset Card 4

With three elements, organize the subsets by size.

F = {banana, apple, lemon}

∅

{banana}, {apple}, {lemon}

{banana,apple}, {banana,lemon}, {apple,lemon}

{banana,apple,lemon}

Power set

The power set is the set of all subsets of a set. Here there are 8 subsets altogether.

Your turn: list all subsets of G={1,2,3}.

List them the way you normally write subsets. Commas or semicolons between subsets are fine. Include ∅ and the whole set.

Subset Card 5

Now the 2ⁿ rule has something to count.

From the lists we built:

1 element

2 subsets

2 elements

4 subsets

3 elements

8 subsets

The pattern

Number of subsets = 2ⁿ

n is the number of elements in the original set.

P={2,4,6,8,10}

Count the elements: n=5.

Use 2⁵.

ANSWER

32 subsets

Q={m,n,p,r}. How many subsets?

Subset Card 6

The whole set is a subset too—but it has a special name.

A = {1,2,3}

Proper subset

A subset that is not equal to the whole set.

{1,2} ⊂ A

Improper subset

The set itself is its improper subset.

A ⊆ A

Examples of proper subsets of A include ∅, {1}, {2}, {3}, {1,2}, {1,3}, and {2,3}.

The whole set {1,2,3} is the improper subset.

Counting proper subsets

If a set has n elements, it has 2ⁿ subsets altogether. Exactly one of those is the whole set itself, so the number of proper subsets is 2ⁿ−1.

For A={1,2,3}: 2³−1=7 proper subsets.

A set has 4 elements. How many proper subsets?

Which is the improper subset of A?

Set Card 7

Cardinality simply means: how many elements?

A={2,4,6,8}

There are four distinct elements.

n(A)=4 or |A|=4

Do not count repeated writing twice. A set contains distinct elements.

B={red,blue,green}. What is its cardinality?

Set Card 8

A set can have no elements at all.

The empty set or null set has no elements.

∅ or { }

Its cardinality is 0.

E = {whole numbers between 2 and 3}

There are no whole numbers strictly between 2 and 3, so E=∅.

Which set is empty?

Set Card 9

Same elements and same number of elements are not the same idea.

Equal sets

Contain exactly the same elements. Order does not matter.

{1,2,3} = {3,2,1}

Equivalent sets

Have the same cardinality.

{a,b,c} and {4,5,6}

A={1,2,3}, B={3,2,1}. What are they?

Venn Card 1

Before two circles, learn the rectangle.

The Universal Set, U, contains all possible elements under consideration.

U={1,2,3,4} A={2,4}

Where is 3?

Important: “Not in A” does not mean “outside the rectangle.” The rectangle tells us what universe we are working in.

Venn Card 2

Why do the circles overlap?

Art Club = {Ana, Ben, Cara}
Music Club = {Ben, Cara, Diego}

Think before the symbol

Ben and Cara belong to both clubs. We need a place that is inside both circles at the same time.

That shared middle is called the intersection. Now the symbol has a meaning: A ∩ B means the elements in A and B.

Venn Card 3

BOTH first. Every time.

A={1,2,3,4} B={3,4,5,6}

Worked example

Find what appears in both: 3 and 4.

Put 3 and 4 in the overlap first.

Put 1,2 in A-only and 5,6 in B-only.

X={2,4,6,8}, Y={4,8,10}. Draw two circles. Put the BOTH numbers first, then finish the diagram.

Both: 4,8 X-only: 2,6 Y-only: 10

Operation Card 1

Intersection means BOTH.

A ∩ B = elements in A AND B

Example 1

A={2,4,6,8,10}, B={3,6,9}

LOOK

Which element appears in both lists?

ANSWER

A∩B={6}

Example 2 — your turn: X={1,3,5,7}, Y={2,3,6,7}

Example 3 — what if nothing is shared? P={1,3,5}, Q={2,4,6}

Tap ∅ if you need the empty-set symbol.

Operation Card 2

Union means everything in A OR B.

A ∪ B = elements in A, B, or both

Worked example

A={1,2,3,4}, B={3,4,5,6}

Combine both lists, but write each distinct element once.

A∪B={1,2,3,4,5,6}

Why aren't 3 and 4 written twice? A set cares whether an element belongs. Repeating an element does not create a new element.

X={2,4,6}, Y={4,6,8}. Find X∪Y.

Venn Card 4

The picture stays. The question changes.

A∩B?

A∪B?

The sets did not change. The operation tells you which part of the same picture to read.

Operation Card 3

Difference: start in the first set.

Teacher Abi shortcut

A−B: Stand inside A. Erase anything that is also in B.

A={a,b,c,d,e} B={c,d,e,f,g}

Start with A: a,b,c,d,e.

Remove c,d,e because they are also in B.

Answer

A−B={a,b}

Now reverse it. What is B−A?

Operation Card 4

Complement needs the Universal Set.

U={1,2,3,4,5,6,7,8,9,10}
A={2,4,6,8,10}

Think

A′ means: look through U and keep everything that is not in A.

REMOVE

2,4,6,8,10

KEEP

1,3,5,7,9

ANSWER

A′={1,3,5,7,9}

Which statement is always true?

Venn Card 5

Four places. Know what each one means.

A-only

A−B

Both

A∩B

B-only

B−A

All three inside pieces

A∪B

Outside both

(A∪B)′

Venn Card 6

Can you translate the picture backward?

Symbols → picture is one skill. Picture → symbols is another.

What does the shaded region represent?

Counting Sets

The bars |A| mean “how many?”

X={2,3,5,7,11,13}

There are six elements, so n(X)=6. You may also see |X|=6; both mean “the number of elements in X.”

For two sets

|A∪B| = |A| + |B| − |A∩B|

Why subtract? Because |A|+|B| counted the BOTH group twice.

|A|=12, |B|=9, |A∩B|=4. Find |A∪B|.

Word Problem Card 1

Four words can completely change the answer.

both

A∩B

A only

A−B

either A or B

A∪B

neither

outside both

Worked example

25 students like Math, 20 like Science, and 10 like both.

ONLY

Math only = 25−10 = 15

EITHER

25+20−10 = 35

If the class has 40 students, how many like neither?

Word Problem Card 2

Translate the story before calculating.

50 students were surveyed. 32 play Basketball, 22 play Volleyball, and every student plays at least one of the two sports. How many play both?

Worked example

“Every student plays at least one” → |B∪V|=50.

50=32+22−|B∩V|

54−4=50, so |B∩V|=4.

New example: 38 use App T, 28 use App I, and 12 use both. How many use I only?

Level Up

Parentheses tell you what to do first.

U={1,2,3,4,5} A={1,2,3} B={3,4}
Find (A∪B)′

Worked example

Do the parentheses: A∪B={1,2,3,4}.

Take the complement inside U. Only 5 is outside the union.

Answer

(A∪B)′={5}

U={1,2,3,4,5,6}, X={1,2,3}, Y={3,4}. Find (X∪Y)′.

When you get stuck

Don’t restart the whole chapter. Find the exact snag.

I don’t know what the subsets are.
Start with ∅. Then list one-element groups, two-element groups, and so on. Finish with the whole set itself.

I don’t know where the numbers go.
Find BOTH first. Put shared elements in the overlap before anything else.

I mix up ∪ and ∩.
Ask: “everything or only shared?” Everything → union. Shared → intersection.

I don’t understand A−B.
Stand inside A. Erase anything also in B.

I don’t understand A′.
Look at U. Keep everything in U that is not in A.

I double-count word problems.
The overlap belongs to both sets, so |A|+|B| counts it twice. Subtract it once.

Practice Studio

Choose how you want to practise.

The goal is not to finish questions. The goal is to notice which idea still breaks down.

Practice Workspace

Fresh practice

Mastery Check

Can you use the ideas when they are mixed?

Five mixed items. Wrong answers give specific feedback. Correct them and check again.

1. A={3,6,9,12}, B={2,3,6,10}. Find A∩B.

2. A={2,5,8}, B={1,5,7,8}. Find A∪B.

3. U={1,2,3,4,5,6,7,8}, A={1,3,6,8}. Find A′.

4. R={a,b,c,d}. How many subsets does R have?

5. 27 students like Math, 19 like Science, and 7 like both. How many like Math only?

Find the Math-only region.

Notebook Updated

I finished this.

Notebook Pocket

★

TEACHER ABI

SET
SOLVER

STUDY NOTEBOOK

MASTERED

Today I can...

• tell elements from subsets
• list all subsets and count them using 2ⁿ
• build and read two-set Venn diagrams
• solve union, intersection, difference, and complement
• use only, both, either, and neither in word problems

Teacher Abi Review Session

Still not understanding it?

That is okay.

Sometimes the symbols make sense but the Venn diagram does not. Sometimes the picture makes sense but the operation does not. That tells us exactly where to rebuild.

In a review session, we can identify the specific step causing the problem and practise from that level upward.

Book a Review Session with Teacher Abi

Choose what you need next.

If one idea still feels shaky, you do not have to redo the whole notebook. Use the chapter menu to jump back to the concept you want to review, then work forward again from there.

Wednesday, September 30, 2026