Operations on 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 1.4.

~5 min read

Summary

Union (\( \cup \)) joins everything that is in A or in B.
Intersection (\( \cap \)) keeps only what is in both A and B.
Complement (\( A’ \)) is everything in the universal set U that is not in A.
De Morgan’s law flips the sign under a complement: \( (A \cup B)’ = A’ \cap B’ \) and \( (A \cap B)’ = A’ \cup B’ \).

Key Words

The union \( A \cup B \) is the set of all elements that are in A or in B, or in both.
The intersection \( A \cap B \) is the set of all elements that are in both A and B.
Two sets are disjoint when they share no elements, so \( A \cap B = \varnothing \).
The complement \( A’ \) is the set of all elements of U that are not in A.
The difference \( A – B \) is the set of elements that are in A but not in B.
The symmetric difference \( A\,\triangle\,B \) is the set of elements that are in A or B but not in both.
The Cartesian product \( A \times B \) is the set of all ordered pairs \( (a, b) \) with \( a \in A \) and \( b \in B \).

The operations at a glance

The shaded part shows the result of each operation.

Union and Intersection

The union \( A \cup B \) takes every element that is in A or in B, or in both. In symbols, \( A \cup B = \{x \mid x \in A \text{ or } x \in B\} \).
The intersection \( A \cap B \) keeps only the elements that are in both sets. In symbols, \( A \cap B = \{x \mid x \in A \text{ and } x \in B\} \).
When two sets share nothing, they are disjoint, and \( A \cap B = \varnothing \).

Complement

The complement \( A’ \) is everything inside the universal set U that is not in A.
An easy fact to remember is that \( A’ = U – A \).

De Morgan’s Law

The complement of a union is \( (A \cup B)’ = A’ \cap B’ \).
The complement of an intersection is \( (A \cap B)’ = A’ \cup B’ \).
In words, when you take the complement of the whole thing, you complement each set and switch the operation, so \( \cup \) becomes \( \cap \) and \( \cap \) becomes \( \cup \).

Difference and Symmetric Difference

The difference \( A – B \) (also written \( A \setminus B \)) is the set of elements that are in A but not in B. You can also find it with \( A – B = A \cap B’ \).
The symmetric difference \( A\,\triangle\,B \) is everything that is in A or in B but not in both. In symbols, \( A\,\triangle\,B = (A – B) \cup (B – A) = (A \cup B) – (A \cap B) \).

Cartesian Product

The Cartesian product \( A \times B \) is the set of all ordered pairs \( (a, b) \) where \( a \) comes from A and \( b \) comes from B.
Order matters, so \( A \times B \) is usually not the same as \( B \times A \).
If A has \( m \) elements and B has \( n \) elements, then \( A \times B \) has \( m \times n \) ordered pairs.

Quick Comparison Table

Using A = {1, 2, 3} and B = {3, 4, 5}

Operation	In words	Result
Union \( A \cup B \)	In A or B (or both)	{1, 2, 3, 4, 5}
Intersection \( A \cap B \)	In both A and B	{3}
Difference \( A – B \)	In A but not in B	{1, 2}
Symmetric difference \( A\,\triangle\,B \)	In A or B but not both	{1, 2, 4, 5}

Common Mistakes to Avoid

Do not mix up union, which means “or” and takes everything, with intersection, which means “and” and keeps only the shared part.
In De Morgan’s law, remember to switch the operation, so \( (A \cap B)’ = A’ \cup B’ \), not \( A’ \cap B’ \).
The difference is not symmetric, so \( A – B \) is usually not the same as \( B – A \).
The Cartesian product depends on order, so \( A \times B \) is usually not the same as \( B \times A \).

Easy Ways to Remember

The union sign \( \cup \) looks like a cup, and a cup holds everything you pour into it.
For De Morgan’s law, think “break the bracket, switch the sign”: the complement goes onto each set and \( \cup \) swaps with \( \cap \).
For the difference \( A – B \), start with A and take away anything that is also in B.

Quiz

Tap an answer to check it. For all three questions, A = {1, 2, 3} and B = {3, 4, 5}.

1. What is \( A \cap B \) (the intersection)?

2. What is \( A – B \) (the difference)?

3. What is the symmetric difference \( A\,\triangle\,B \)?

Remember: Set operations build new sets from old ones. Union joins, intersection overlaps, complement is the outside, difference subtracts, and De Morgan's law switches \( \cup \) and \( \cap \) under a complement.