Sets
In maths, a set is a gathering of particular articles, considered as an item in its own right. For instance, the numbers 2, 4, and 6 are particular articles when considered independently, however when they are considered on the whole they structure a solitary arrangement of size three, composed {2, 4, 6}. The idea of a set is one of the most major in arithmetic. Created toward the finish of the nineteenth century, set hypothesis is presently an omnipresent piece of arithmetic, and can be utilized as an establishment from which almost all of science can be determined. In science instruction, basic themes from set hypothesis, for example, Venn graphs are educated at a youthful age, while further developed ideas are instructed as a component of a college degree.
If every element of set A is also in B, then A is said to be a subset of B, written A ⊆ B (pronounced A is contained in B). Equivalently, one can write B ⊇ A, read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by ⊆ is called inclusion or containment, and is given also for equal sets, that is, equality of sets is the same as mutual containment in each other: A ⊆ B and B ⊆ A is equivalent to A = B.
If A is a subset of B, but not equal to B, then A is called a proper subset of B, written A ⊊ B, or simply A ⊂ B (A is a proper subset of B), or B ⊋ A (B is a proper superset of A, B ⊃ A).
The expressions A ⊂ B and B ⊃ A are used differently by different authors; some authors use them to mean the same as A ⊆ B (respectively B ⊇ A), whereas others use them to mean the same as A ⊊ B (respectively B ⊋ A).
A is a subset of BExamples:
The set of all men is a proper subset of the set of all people.{1, 3} ⊆ {1, 2, 3, 4}.{1, 2, 3, 4} ⊆ {1, 2, 3, 4}.There is a unique set with no members, called the empty set (or the null set), which is denoted by the symbol ∅ (other notations are used; see empty set). The empty set is a subset of every set, and every set is a subset of itself:
∅ ⊆ A.A ⊆ A.The above characterization of set equality can be used to show that two sets described differently are, in fact, equal:
A = B if and only if A ⊆ B and B ⊆ A.A partition of a set S is a set of nonempty subsets of S such that every element x in S is in exactly one of these subsets, that is, any two sets of the partition contain no element in common, they are said to be disjoint, and the union of all elements of the partition that are sets themselves, make up S.
Power setsMain article: Power setThe power set of a set S is the set of all subsets of S. The power set contains S itself and the empty set because these are both subsets of S. For example, the power set of the set {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The power set of a set S is usually written as P(S).
The power set of a finite set with n elements has 2n elements. For example, the set {1, 2, 3} contains three elements, and the power set shown above contains 23 = 8 elements.
The power set of an infinite (either countable or uncountable) set is always uncountable. Moreover, the power set of a set is always strictly "bigger" than the original set in the sense that there is no way to pair every element of S with exactly one element of P(S). (There is never an onto map or surjection from S onto P(S).)
Every partition of a set S is a subset of the powerset of S.
CardinalityMain article: CardinalityThe cardinality of a set S, denoted |S|, is the number of members of S. For example, if B = {blue, white, red}, then |B| = 3. Repeated members in an extensional definition are not counted, so |{blue, white, red, blue, white}| = 3, too.
The cardinality of the empty set is zero. For example, the set of all three-sided squares has zero members and thus is the empty set. Though it may seem trivial, the empty set, like the number zero, is important in mathematics. Indeed, the existence of this set is one of the fundamental concepts of axiomatic set theory.
Some sets have infinite cardinality. The set N of natural numbers, for instance, is infinite. Some infinite cardinalities are greater than others. For instance, the set of real numbers has greater cardinality than the set of natural numbers. However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.
Special sets
The natural numbers ℕ are contained in the integers ℤ, which are contained in the rational numbers ℚ, which are contained in the real numbers ℝ, which are contained in the complex numbers ℂThere are some sets or kinds of sets that hold great mathematical importance and are referred to with such regularity that they have acquired special names and notational conventions to identify them. One of these is the empty set, denoted { } or ∅. A set with exactly one element, x, is a unit set, or singleton, {x}.[2]
Many of these sets are represented using blackboard bold or bold typeface. Special sets of numbers include
P or ℙ, denoting the set of all primes: P = {2, 3, 5, 7, 11, 13, 17, ...}.N or {\displaystyle \mathbb {N} } \mathbb {N} , denoting the set of all natural numbers: N = {0, 1, 2, 3, ...} (sometimes defined excluding 0).Z or {\displaystyle \mathbb {Z} } \mathbb {Z} , denoting the set of all integers (whether positive, negative or zero): Z = {..., −2, −1, 0, 1, 2, ...}.Q or ℚ, denoting the set of all rational numbers (that is, the set of all proper and improper fractions): Q = {a/b | a, b ∈ Z, b ≠ 0}. For example, 1/4 ∈ Q and 11/6 ∈ Q. All integers are in this set since every integer a can be expressed as the fraction a/1 (Z ⊊ Q).
If A is a subset of B, but not equal to B, then A is called a proper subset of B, written A ⊊ B, or simply A ⊂ B (A is a proper subset of B), or B ⊋ A (B is a proper superset of A, B ⊃ A).
The expressions A ⊂ B and B ⊃ A are used differently by different authors; some authors use them to mean the same as A ⊆ B (respectively B ⊇ A), whereas others use them to mean the same as A ⊊ B (respectively B ⊋ A).
A is a subset of BExamples:
The set of all men is a proper subset of the set of all people.{1, 3} ⊆ {1, 2, 3, 4}.{1, 2, 3, 4} ⊆ {1, 2, 3, 4}.There is a unique set with no members, called the empty set (or the null set), which is denoted by the symbol ∅ (other notations are used; see empty set). The empty set is a subset of every set, and every set is a subset of itself:
∅ ⊆ A.A ⊆ A.The above characterization of set equality can be used to show that two sets described differently are, in fact, equal:
A = B if and only if A ⊆ B and B ⊆ A.A partition of a set S is a set of nonempty subsets of S such that every element x in S is in exactly one of these subsets, that is, any two sets of the partition contain no element in common, they are said to be disjoint, and the union of all elements of the partition that are sets themselves, make up S.
Power setsMain article: Power setThe power set of a set S is the set of all subsets of S. The power set contains S itself and the empty set because these are both subsets of S. For example, the power set of the set {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The power set of a set S is usually written as P(S).
The power set of a finite set with n elements has 2n elements. For example, the set {1, 2, 3} contains three elements, and the power set shown above contains 23 = 8 elements.
The power set of an infinite (either countable or uncountable) set is always uncountable. Moreover, the power set of a set is always strictly "bigger" than the original set in the sense that there is no way to pair every element of S with exactly one element of P(S). (There is never an onto map or surjection from S onto P(S).)
Every partition of a set S is a subset of the powerset of S.
CardinalityMain article: CardinalityThe cardinality of a set S, denoted |S|, is the number of members of S. For example, if B = {blue, white, red}, then |B| = 3. Repeated members in an extensional definition are not counted, so |{blue, white, red, blue, white}| = 3, too.
The cardinality of the empty set is zero. For example, the set of all three-sided squares has zero members and thus is the empty set. Though it may seem trivial, the empty set, like the number zero, is important in mathematics. Indeed, the existence of this set is one of the fundamental concepts of axiomatic set theory.
Some sets have infinite cardinality. The set N of natural numbers, for instance, is infinite. Some infinite cardinalities are greater than others. For instance, the set of real numbers has greater cardinality than the set of natural numbers. However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.
Special sets
The natural numbers ℕ are contained in the integers ℤ, which are contained in the rational numbers ℚ, which are contained in the real numbers ℝ, which are contained in the complex numbers ℂThere are some sets or kinds of sets that hold great mathematical importance and are referred to with such regularity that they have acquired special names and notational conventions to identify them. One of these is the empty set, denoted { } or ∅. A set with exactly one element, x, is a unit set, or singleton, {x}.[2]
Many of these sets are represented using blackboard bold or bold typeface. Special sets of numbers include
P or ℙ, denoting the set of all primes: P = {2, 3, 5, 7, 11, 13, 17, ...}.N or {\displaystyle \mathbb {N} } \mathbb {N} , denoting the set of all natural numbers: N = {0, 1, 2, 3, ...} (sometimes defined excluding 0).Z or {\displaystyle \mathbb {Z} } \mathbb {Z} , denoting the set of all integers (whether positive, negative or zero): Z = {..., −2, −1, 0, 1, 2, ...}.Q or ℚ, denoting the set of all rational numbers (that is, the set of all proper and improper fractions): Q = {a/b | a, b ∈ Z, b ≠ 0}. For example, 1/4 ∈ Q and 11/6 ∈ Q. All integers are in this set since every integer a can be expressed as the fraction a/1 (Z ⊊ Q).
R or {\display
The German word Menge, rendered as "set" in English, was begat by Bernard Bolzano in his work The Paradoxes of the Infinite.
Definition
A set is a well-characterized gathering of particular items. The items that make up a set (otherwise called the set's components or individuals) can be anything: numbers, individuals, letters of the letters in order, different sets, etc. Georg Cantor, one of the authors of set hypothesis, gave the accompanying meaning of a set toward the start of his Beiträge zur Begründung der transfiniten Mengenlehre:[1]
A set is an assembling into an entire of unequivocal, particular objects of our observation [Anschauung] or of our idea—which are called components of the set.
Sets are traditionally meant with capital letters. Sets An and B are equivalent if and just on the off chance that they have correctly the equivalent elements.[2]
For specialized reasons, Cantor's definition ended up being insufficient; today, in settings where more thoroughness is required, one can utilize aphoristic set hypothesis, in which the thought of a "set" is taken as a crude idea and the properties of sets are characterized by a gathering of sayings. The most essential properties are that a set can have components, and that two sets are equivalent (one and the equivalent) if and just if each component of each set is a component of the other; this property is known as the extensionality of set