The Notion of Sets

These QuickNotes give you the most important points to remember for the ESSLCE. They are based on the MoE Grade 9 Mathematics Textbook, Unit 1, Chapter 1.3.

~5 min read

Summary

An empty set has no elements. We write it { } or ∅.
A finite set has a fixed number of elements; an infinite set never ends.
Equal sets have exactly the same elements. Equivalent sets have the same number of elements.
A ⊆ B means every element of A is also in B. A set with n elements has 2ⁿ subsets.

Key Words

An empty set (null set) has no elements, written { } or ∅.
A finite set has a fixed number of elements you can count.
An infinite set has elements that never end.
n(A) is the number of elements in set A.
Equal sets (A = B) have exactly the same elements.
Equivalent sets (A ↔ B) have the same number of elements.
The universal set U holds all the elements of the sets being discussed.
A is a subset of B (A ⊆ B) if every element of A is also in B.
A is a proper subset of B (A ⊂ B) if A ⊆ B and A ≠ B.

Empty, Finite, and Infinite Sets

An empty set has no elements at all. We write it as { } or ∅. For example, there is no natural number between 1 and 2, so that set is empty.
A finite set has a fixed number of elements you can count, like {1, 2, 3, …, 10}. An infinite set never ends, like the whole numbers {0, 1, 2, 3, …}.
We write n(A) for the number of elements in A. For A = {1, 2, 3, …, 10}, n(A) = 10.

Equal Sets and Equivalent Sets

Two sets are equal (A = B) when they have exactly the same elements. The order does not matter, so {1, 2, 3, 4} = {4, 3, 2, 1}.
Two sets are equivalent (A ↔ B) when they have the same number of elements, even if the elements are different. So {1, 2, 3} and {a, b, c} are equivalent because each has 3 elements.
Equal sets are always equivalent, but equivalent sets are not always equal.

Equal vs Equivalent

Feature	Equal (A = B)	Equivalent (A ↔ B)
What must match	The elements themselves	Only the number of elements
Elements	Exactly the same	Can be different
Example	{1, 2, 3} = {3, 2, 1}	{1, 2, 3} ↔ {a, b, c}

Universal Set

The universal set U holds all the elements of the sets you are working with. If A is the even numbers and B is the odd numbers, then U can be all the natural numbers.

Subset and Proper Subset

Every element of A is also in B, so A ⊆ B.

A is a subset of B (A ⊆ B) when every element of A is also in B. For example, {1, 2, 3} ⊆ {1, 2, 3, 4}.
A is a proper subset of B (A ⊂ B) when A ⊆ B but A ≠ B, which means B has at least one extra element.
Two useful facts: the empty set is a subset of every set, and every set is a subset of itself.
A set with n elements has 2ⁿ subsets and 2ⁿ – 1 proper subsets. So {1, 2, 3} has 2³ = 8 subsets and 7 proper subsets.

Subset vs Proper Subset

Feature	Subset (A ⊆ B)	Proper subset (A ⊂ B)
Meaning	Every element of A is in B	Same, but A ≠ B
Can A equal B?	Yes	No
Count for n elements	2ⁿ	2ⁿ – 1

Common Mistakes to Avoid

Equal is not the same as equivalent. Equal sets have the same elements; equivalent sets only have the same number of elements.
The symbol ∈ is for one element, but ⊆ is for a whole set. Write 1 ∈ {1, 2} but {1} ⊆ {1, 2}.
The empty set is a subset of every set, including itself.
A set is a subset of itself, but it is not a proper subset of itself.

Easy Ways to Remember

Equal means the same members. Equivalent means the same number.
Subsets of a set with n elements: 2ⁿ. Proper subsets: 2ⁿ – 1 (take away the set itself).

Quiz

Tap an answer to check it.

1. Which one of the following is an empty set?

2. The sets A = {1, 2, 3} and B = {a, b, c} are:

3. If A = {1, 2, 3}, the total number of subsets of A is:

4. Which one of the following statements is NOT correct?

Remember: Equal sets share the same elements, equivalent sets share the same number of elements, and A ⊆ B means every element of A is also in B. A set with n elements has 2ⁿ subsets.