Sets & Relations

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Explore sets, set operations, Venn diagrams, relations, and functions - the language of modern mathematics.

Concepts in this topic

Set Basics & Operations - Understand sets, membership, subsets, and fundamental set operations
Venn Diagrams & Properties - Visualize set operations and learn key set identities
Relations - Learn about binary relations and their properties
Functions - Understand functions as special relations, injections, surjections, and bijections

Key formulas and identities for Sets & Relations, with each formula and what it means.
Name	Formula	Meaning
Union	`A ∪ B = {x \| x ∈ A or x ∈ B}`	The set of elements belonging to at least one of the two sets
Intersection	`A ∩ B = {x \| x ∈ A and x ∈ B}`	The set of elements belonging to both sets
Complement	`Aᶜ = {x ∈ U \| x ∉ A}`	The set of elements in the universal set that are not in A
De Morgan's Law for Sets (1)	`(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ`	The complement of a union is the intersection of the complements
De Morgan's Law for Sets (2)	`(A ∩ B)ᶜ = Aᶜ ∪ Bᶜ`	The complement of an intersection is the union of the complements
Inclusion-Exclusion (Two Sets)	`\|A ∪ B\| = \|A\| + \|B\| - \|A ∩ B\|`	The size of a union equals the sum of sizes minus the intersection
Power Set Size	`\|P(A)\| = 2ⁿ (n = \|A\|)`	A set with n elements has 2ⁿ subsets
Cartesian Product Size	`\|A × B\| = \|A\| × \|B\|`	The number of ordered pairs is the product of the set sizes

From the free practice sample - 76 guided problems cover this topic in interactive practice.

If A = {1, 2, 3} and B = {3, 4, 5}, what is A ∩ B?

Show answer

Answer: {3}

A ∩ B is the set of elements in both A and B. The only element in both is 3, so A ∩ B = {3}.