Finite and Infinite Sets

Lesson 1 of 4: Sets

In this lesson:

• Decide whether a set can be completely listed
• Use set notation and find the cardinal number
• Recognise the empty set ∅

Learning Objectives for Today's Lesson

You should be able to:

1. Define finite set and infinite set
2. List members using curly-brace notation
3. Find the cardinal number of a finite set
4. Classify a set as finite, infinite, or empty
5. Explain why a large collection is not infinite

Can You Finish the List?

Two questions — think before you answer:

1. List every student sitting in your classroom right now.
2. List every whole number starting from 1.

Which one can you actually finish?

The Fruit Bowl: Listing Three Members

A bowl holds three fruits: mango, banana, orange.

List all members: {mango, banana, orange}

• Start counting: mango… banana… orange… done.
• Three members. You finished.

This is what "finite" feels like.

Defining the Finite Set Formally

A finite set is a collection that can be completely listed — listing eventually stops.

Every member can be counted and the count has a definite end.

Set Notation and the Cardinal Number

A set is written with curly braces { }, members separated by commas.

• = {weekdays} = {Mon, Tue, Wed, Thu, Fri}
• ← the cardinal number (member count)
• Each item is a member (or element)

Membership: "" — Monday is a member.

Worked Example: Set Notation for Letters

Set = {letters in the word UGANDA}

Step 1 — list: U, G, A, N, D, A — A repeats; sets do not repeat.

Step 2 — remove duplicate: = {U, G, A, N, D}

Step 3 — count:

Is finite? Yes.

Quick Check: Classify the Letter Set

Set = {letters in the word KAMPALA}

• List all distinct letters.
• Write in set notation.
• Find .
• Is finite or infinite?

Try it yourself before the next slide.

From Finite Lists to Infinite Ones

You finished listing the letters in KAMPALA.

Now try this: list every whole number starting from 1.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 …

Will you ever stop?

Defining the Infinite Set Formally

An infinite set is a collection whose listing never terminates — no matter how long you list, there is always another member.

We write: = {1, 2, 3, 4, 5, …} — the three dots (ellipsis) signal "continues forever."

Key Infinite Sets Worth Knowing

Three infinite sets worth knowing:

• = {1, 2, 3, 4, …} — counting numbers (naturals)
• = {…, −3, −2, −1, 0, 1, 2, 3, …} — all integers
• Even numbers = {2, 4, 6, 8, …} — every second counting number

Each never ends.

Worked Example: An Infinite Multiples Set

Set = {multiples of 3}

= {3, 6, 9, 12, 15, …}

• Listing starts: 3, 6, 9, 12 …
• After any term , the next is — never ends.
• is infinite; is not a whole number.

The "…" signals an unfinished list.

Predict First: The Jar of Sand

A jar holds one million grains of sand.

Is this set of grains infinite?

• A. Yes — one million is so large it might as well be infinite.
• B. No — it's a very large set, but not infinite.

Why Large Is Not the Same as Infinite

Test: Can you name the last member?

A last member means finite, however enormous.

Quick Check — Classify These

Which are finite and which are infinite?

1. {even numbers between 1 and 20}
2. {all students at your school}
3. {all counting numbers greater than 1 000}
4. {prime numbers}

Apply the "can you name the last member?" test.

What If There Are No Members?

Can a collection have zero members?

For example: {even numbers between 4 and 6}

The only whole number strictly between 4 and 6 is 5, and 5 is odd, not even. So there are no even numbers between 4 and 6 — the set has no members.

The Empty Set and Its Notation

A set with no members is called the empty set.

• Written as ∅ or { } — both mean the same thing
• — the cardinal number is 0
• The empty set is finite (listing it stops immediately — there is nothing to list)

Find the Error in This Classification

One entry is wrong — find it.

Which is wrong, and why?

Worked Example: Full Set Classification

Classify each set and justify:

Your Turn — Classify Five Sets

Classify each as finite, infinite, or empty (∅):

1. {odd numbers}
2. {whole numbers less than 10}
3. {triangles with 4 sides}
4. {multiples of 5}
5. {students in your family who are also teachers}

Write your answers before advancing.

Watch Out — Common Errors

✓ A large set is finite with a last member
✓ The empty set ∅ is valid — no members
✓ means zero members

Error 1: {grains of sand} is finite
Error 2: the empty set is real
Error 3:

Key Takeaways from Today's Lesson

✓ A finite set can be completely listed — it has a cardinal number
✓ An infinite set never terminates — the ellipsis … signals this
✓ The empty set ∅ has zero members — it's finite
✓ Large ≠ infinite — ask: "Name the last member?"

Next: counting subsets.