Cartesian Products
All ordered pairs from two sets.
Key idea
All ordered pairs from two sets.
In depth
A × B is the set of all ordered pairs (a, b) with a ∈ A, b ∈ B. Its size is |A||B|. It models grids, coordinates and combinations of choices.
Worked example
{1, 2} × {x, y} = {(1, x), (1, y), (2, x), (2, y)}.
Formula
Real-world uses
- Listing all combinations of shirt sizes and colours
- Building all possible game board positions
Common mistake
Assuming A × B = B × A; the order inside pairs matters.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Cartesian Products?
All ordered pairs from two sets. It is a topic at the intermediate level in Discrete Math & Number Theory.
What is the formula for Cartesian Products?
The key formula is |A×B| = |A||B|.
Where is Cartesian Products used in real life?
Real-world examples: Listing all combinations of shirt sizes and colours; Building all possible game board positions.
What should I learn before Cartesian Products?
Make sure you are comfortable with Inclusion–Exclusion Principle, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Cartesian Products?
Assuming A × B = B × A; the order inside pairs matters.
How can I practise Cartesian Products?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Cartesian Products?
Move on to Relations next.
Related intermediate topics in Discrete Math & Number Theory
- Statements and Truth Values – A proposition is true or false, never both.
- Logical Connectives – AND, OR, NOT, implication and biconditional.
- Truth Tables – List all truth assignments to evaluate compound statements.
- Subsets and Power Sets – A set with n elements has 2ⁿ subsets.
- Set Operations – Union, intersection, difference and complement.
- Venn Diagrams – Pictures of sets and their overlaps.