Aleph numberIn mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ).
Axiom of choiceIn mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, one can identify another set containing one element chosen from each set, even if the collection is infinite.
Cardinal numberA bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4.Aleph-null, the smallest infinite cardinalIn mathematics, a cardinal number, or cardinal for short, is a kind of number that measures the cardinality of a set, i.e., how many elements there are in a set.Cardinality of the continuumIn set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers R {\displaystyle \mathbb {R} } , sometimes called the continuum. It is an infinite cardinal number and is denoted by c {\displaystyle {\mathbf {\mathfrak {c}}}} (lowercase Fraktur "c") or | R | .
Injective functionIn mathematics, an injective function (also known as injection, or one-to-one function) is a functionf that maps distinct elements of its domain to distinct elements of its codomain; that is, x1 ≠ x2 implies f(x1) ≠ f(x2) (equivalently by contraposition, f(x1) = f(x2) implies x1 = x2). In other words, every element of the function's codomain is the image of at most one element of its domain.
SequenceIn mathematics, a sequence is a collection of objects possibly with repetition, that come in a specified order. Like a set, it contains members (also called elements, or terms). Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. The notion of a sequence can be generalized to an indexed family, defined as a function from an arbitrary index set.
Real numberIn mathematics, a real number is a number that can be used to measure a continuous one-dimensionalquantity such as a length, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion.
Infinite setIn set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. PropertiesThe set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC), but only by showing that it follows from the existence of the natural numbers.
Natural numberNatural numbers can be used for counting: one apple plus two apples equals three apples.In mathematics, the natural numbers are the numbers used for counting up, starting from 0 or 1. The first few natural numbers are 0 (if included), 1, 2, 3, and so on. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set of the natural numbers is commonly denoted by a bold N or a blackboard boldN{\displaystyle \mathbb {N} }.
Countable setMathematical set that can be enumeratedA mathematical set is countable if either it is finite or it can be put in one to one correspondence with the set of natural numbers.
Element of a setIn mathematics, an element (or member) of a set is any one of the distinctobjects that belong to that set. For example, given a set called A containing the first four positive integers(A={1,2,3,4}{\displaystyle A=\{1,2,3,4\}}), one could say that "3 is an element of A", expressed notationally as 3∈A{\displaystyle 3\in A}. SetsWriting A={1,2,3,4}{\displaystyle A=\{1,2,3,4\}} means that the elements of the set A are the numbers 1, 2, 3 and 4.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.*As an Amazon Associate I earn from qualifying purchases.