Explore set theory with interactive Venn diagrams and comprehensive set operations. Visualize set relationships and perform complex calculations.
Elements in A or B or both
Elements in both A and B
Elements in A but not in B
Elements in U but not in A
Elements in A or B but not both
Ordered pairs (a,b) where a∈A, b∈B
Set theory is a branch of mathematical logic that studies collections of objects, called sets. It forms the foundation for many areas of mathematics.
A well-defined collection of distinct objects, called elements or members.
Example: A = {1, 2, 3, 4}
An object that belongs to a set. We write a ∈ A to mean "a is an element of A".
Example: 2 ∈ {1, 2, 3}
The set containing no elements, denoted as ∅ or {}.
Example: ∅ = {}
The set that contains all elements under consideration, denoted as U.
Example: U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
The union of sets A and B contains all elements that are in A, or in B, or in both.
Notation: A ∪ B = {x : x ∈ A or x ∈ B}
Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}
The intersection of sets A and B contains all elements that are in both A and B.
Notation: A ∩ B = {x : x ∈ A and x ∈ B}
Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B = {3}
The difference A - B contains all elements that are in A but not in B.
Notation: A - B = {x : x ∈ A and x ∉ B}
Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A - B = {1, 2}
The complement of set A contains all elements in the universal set U that are not in A.
Notation: A' = {x : x ∈ U and x ∉ A}
Example: If U = {1, 2, 3, 4, 5} and A = {1, 2}, then A' = {3, 4, 5}
The symmetric difference contains elements that are in either A or B, but not in both.
Notation: A △ B = (A - B) ∪ (B - A)
Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A △ B = {1, 2, 4, 5}
