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The number of elements in a set. Example: The cardinality of {1, 2, 3} is 3.

A collection of distinct objects, considered as an object in its own right. Example: The set of natural numbers.

A relation from a set of inputs to a set of possible outputs where each input is assigned to exactly one output. Example: f(x) = x^2 maps any x to its square.

A set A is a subset of a set B if every element of A is also an element of B. Example: {2, 4} is a subset of the set of even numbers.

The set of all actual outputs (y-values) that a function can produce. Example: The range of f(x) = x^2 is all nonnegative real numbers.

A set with an unlimited number of elements. Example: The set of all natural numbers is infinite.

The complement of a set A refers to elements not in A, typically within a universal set U. Example: If U = {1, 2, 3, 4} and A = {1, 2}, then the complement of A is {3, 4}.

The set of all possible inputs (x-values) for which a function is defined. Example: The domain of f(x) = 1/x is all real numbers except for x=0.

A function is bijective if every element of the domain is mapped to exactly one unique element of the codomain, and each element of the codomain is the image of exactly one element of the domain. Example: f(x) = 2x is bijective from the reals to the non-zero reals.

A relation that is reflexive, symmetric, and transitive. Example: Congruence modulo n is an equivalence relation.

A way of dividing a set into disjoint subsets such that every element of the set is included in exactly one subset. Example: The set {1, 2, 3, 4} can be partitioned into {{1, 2}, {3, 4}}.

The set of all ordered pairs where the first element is from set A and the second is from set B. Example: If A = {1, 2} and B = {3, 4}, A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}.

A set with no elements. Example: \(\emptyset\) or \({}\).

Two sets A and B are disjoint if they have no elements in common. Example: {1, 2} and {3, 4} are disjoint sets.

Two transformation rules that relate the union and intersection through complementation. Example: \((A \cup B)^c = A^c \cap B^c\) and \((A \cap B)^c = A^c \cup B^c\).

A pair of elements with a defined first and second element. Example: (3, 4) is an ordered pair where 3 is the first element and 4 the second.

A function is injective if it maps distinct elements of the domain to distinct elements of the codomain. Example: f(x) = x^3 is injective because it never maps two different x values to the same y.

An object that is a member of a set. Example: 3 is an element of the set of natural numbers.

A set A is a proper subset of B if A is a subset of B and A is not equal to B. Example: {2, 4} is a proper subset of the natural numbers.

A standard form of axiomatic set theory and the foundation for most of mathematics. Example: ZF includes the axiom of choice indirectly if combined, forming ZFC (Zermelo-Fraenkel with the Axiom of Choice).

The intersection of two sets A and B is the set containing all elements that are both in A and B. Example: The intersection of {1, 2} and {2, 3} is {2}.

The set that contains all objects under consideration for a particular discussion. Example: In a discussion about numbers, U might be the set of all integers.

A set of ordered pairs where the first element is related to the second in some way. Example: The relation of being greater than on natural numbers can be a set of pairs like {(2, 1), (3, 1), ...}.

The power set of a set A is the set of all possible subsets of A. Example: The power set of {1, 2} is {{}, {1}, {2}, {1, 2}}.

A set with a limited number of elements. Example: The set {1, 2, 3, 4, 5} is finite.

The union of two sets A and B is the set containing all elements of A, all elements of B, or both. Example: The union of {1, 2} and {2, 3} is {1, 2, 3}.

A diagram representing mathematical or logical sets pictorially as circles or closed curves, common areas representing intersection. Example: Two overlapping circles showing union and intersection.

The set of all potential outputs for a function, not necessarily all achieved by the function. Example: For f(x) = x^2, the codomain might be all real numbers, even though only nonnegative numbers are achieved.