introduction

Set

A collection of distinct objects, considered as an object in its own right. Example:{1, 2, 3, 4, 5}.

Element

An object that is a member of a set. Example:In {a, b, c}, "a" is an element.

Subset

Set A is a subset of set B if every element of A is also in B. Example:{2, 4} is a subset of {1, 2, 3, 4, 5}.

Union (A ∪ B)

Set of objects in A, B, or both. Example:A ∪ B = {1, 2, 3, 4, 5}.

Intersection (A ∩ B)

Set of objects in both A and B. Example:A ∩ B = {2, 3}.

Difference (A - B)

Set of members in A not in B. Example:A - B = {1, 2}.

Complement

Set of elements not in a given set within a universal set. Example:If A = {2, 3}, A' = {1, 4, 5}.

Empty Set (∅)

Set with no elements, a subset of every set.

Universal Set (U)

Set containing all objects under consideration.

Power Set (P(S))

Set of all subsets of S, including ∅ and S itself.

Associative Law

For sets A, B, C, (A ∪ B) ∪ C = A ∪ (B ∪ C) and (A ∩ B) ∩ C = A ∩ (B ∩ C).

Commutative Law

For sets A, B, A ∪ B = B ∪ A and A ∩ B = B ∩ A.

Distributive Law

For sets A, B, C, A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) and A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

De Morgan's Laws

For sets A, B, (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.

Russell's Paradox

Contradiction in the set of all sets that do not contain themselves.

Ordered Pair

Pair of elements (a, b) where order matters. Example:(1, 2) ≠ (2, 1).

Cartesian Product (A × B)

Set of all ordered pairs (a, b) from A and B.

Relation

Subset of the Cartesian product of two sets, a connection between elements. Example:R = {(1, x), (1, y), (2, x)}.