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Set theory is the core of modern mathematics and serves as a language for mathematicians to discuss and organize their ideas. It is a crucial and elegant concept at its core. However, set by itself is not defined in mathematics because it is a fundamental brick of science that cannot be expressed through other concepts. It is common to give a descriptive explanation of the set.

Sets

9Not a definition) In mathematics, a set is a collection of different objects; they are called elements or members of the set and are typically mathematical objects of any kind: numbers, characters, shapes, symbols, points in space, lines, other geometric shapes, variables, or even other sets.

The way set theory lets us classify, compare, and evaluate these collections is what makes it so powerful. It has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. It is assumed that a set is a well-defined collection of objects. That is, to define a set, we must know for sure whether an element belongs to .it or not. Although the notion of set is not well-defined in wide generality as it leads to paradoxes like Russell’s Paradox, published by Bertrand Russell (1872–1970) in 1901.

If element x belongs to set X, then it is denoted by \( \displaystyle \ x \in X \ \) and if x is not an element of X, then \( \displaystyle \ x \notin X. \quad \) Two sets, X and Y, are said to be equal if they have the same it is denoted by . x ∈ elements.

Let X and Y be any two sets, then X is a subset of Y, denoted by X ⊆ Y, if every element of X is also an element of Y. Two sets, X and Y, are equal if and only if X ⊆ Y and Y ⊆ X.

A set can be defined in several ways. Commonly, a set is defined by either listing all the entries explicitly, called the Roster form, or by stating the properties that are meaningful and unambiguous for elements of the set, called the set builder notation.    

Example 1: Roster or enumeration notation is a notation introduced by the German mathematician Ernst Zermelo in 1908 that specifies a set by listing its elements between braces, separated by commas. For example, one knows that { 4 , 2 , 1 , 3 } and { dollar, euro } denote sets and not tuples because of the enclosing braces. THis notation is used for finite sets withfew elements.

The notation { } for the empty set ∅ is an example of roster notation. A null set, often known as an empty set,

For any positive integer n, the set contating all integers between 1 and n is denoted as [1..n = { 1, 2, 3, … , n }.

Obviously, the roster notation is not suitable for infinite sets. The set-builder notation may help in this case. if ⁠ P(x) is a logical formula depending on a variable ⁠ x⁠, which evaluates to true or false depending on the value of ⁠x, then \[ \left\{ x \ \mid \ P(x) \right\} \] or \[ \left\{ x \ : \ P(x) \right\} \] denotes the set of all ⁠ x⁠ for which ⁠ P (x) ⁠ is true. For example, a set [1..9] can be specified as follows \[ [1..9] = \left\{ n \ \mid \ n \mbox{ is an integer. and } 1 \le x \le 9 \right\} . \] In this notation, the vertical bar "|" is read as "such that", and the whole formula can be read as [1..9] is the set of all n such that n is an integer in the range from 1 to 9 inclusive". On the other hand, closed interval \[ [1, 9] = \left\{ x \in \mathbb{R} \ : \ 1 \le x \le 9 \right\} . \] Here are some familiar collection/sets of numbers.

  • ℕ = { 0, 1, 2, \ldots } is the set of all nonnegative integers;
  • ℤ -- the set of all integers --- { … , −3, −2, −1, 0, 1,2, 3, … };
  • ℚ --- the set of all rational numbers --- { p/q : p, q ∈ ℤ and q ≠ 0 } ;
  • ℝ is the set of all real numbers;
  • ℂ is the set of all complex numbers.
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End of Example 1

Note:    The order presenting elements in a set doesnot matter. For a set, all that matters is whether each element is in it or not; so, the set is not changed if one changes the order or repeat some elements. Therefore, it is a custom to write list of elements without repetition. So, one has, for example,

\[ \left\{ 1, 2, 3 \right\} = \left\{ 3, 2, 1 \right\} = \left\{ 1, 3, 2 \right\} = \left\{ 2, 2, 2, 3, 1 \right\} . \]

Usually, in a particular context, we have to deal with the elements and subsets of a basic set which is relevant to that particular context. This basic set is called the “Universal Set” and is denoted by Ω. It is widely used in probability to represent all possible sample space of an event.

The cardinality of a set X is the number of elements in X. A set X can be finite or infinite depending on the number of elements in X. Cardinality of X is denoted by |X|.
   
Example 2: Let us consider the Latin alphabet that consists of 26 letters. Suppose you write the word "hello." The corresponding set of letters S = { h, e, l, o } has cardinality 4 because it contains four distinct letters.    ■
End of Example 2

 

Set Operations

Set operations are fundamental mathematical tools for constructing, manipulat- ing, and analyzing sets. They enable the combination, comparison, and modification of sets in order to acquire insights and solve various mathematical and real-life problems. Set operations are incorporated in all comuter packges despit that computers always use ordered sets.

There are several standard operations that produce new sets from given sets, in the same way as addition and multiplication produce new numbers from given numbers. Union (combining items from several sets), intersection (finding common elements between sets), complement (identifying elements not in a set), and set difference (removing elements from one set based on another) are the fundamental set operations. We visialize these set operations using Venn diagrams.

The union of two sets ⁠ A ⁠ and ⁠ B is a set denoted ⁠ A ∪ B whose elements are those elements that belong to ⁠ A or ⁠ B or both. That is,

\[ A \cup B = \left\{ x \ : \ x \in A \vee x \in B \right\} , \]
where ∨ denotes the logical or.

RJB https://en.wikipedia.org/wiki/Set_(mathematics)
Union of two sets.

Union is associative and commutative; this means that for proceeding a sequence of intersections, one may proceed in any order, without the need of parentheses for specifying the order of operations. The empty set is an identity element for the union operation.

The intersection of two sets ⁠ A and ⁠ B is a set denoted ⁠ A ∩ B whose elements are those elements that belong to both ⁠ A and ⁠ B. That is,

\[ A \cap B = \left\{ x\ : \ x \in A\ \wedge\ x \in B \right\} , \]
where ∧ denotes the logical and.

Intersection is associative and commutative; this means that for proceeding a sequence of intersections, one may proceed in any order, without the need of parentheses for specifying the order of operations. Intersection has no general identity element. However, if one restricts intersection to the subsets of a given set ⁠ Ω, intersection has ⁠ Ω as identity element.

RJB https://en.wikipedia.org/wiki/Set_(mathematics)
Intersection of two sets.

Let X and Y be two sets. The difference of Y related to X, denoted by X \ Y, is the set of all elements in X which are not in Y. \[ X \setminus Y = \left\{ x \ : \ x \in X \wedge x \notin Y \right\} . \] The difference of a set X related to its universal set Ω is called the complement of X and is denoted by Xc. That is, . \[ X^c = \Omega; \ X = \left\{ x \ : \ x \in \Omega \ \mbox{ and }\ x \notin X \right\} . \] Keep in mind that . Ωc = ∅ and ∅c = Ω .
RJB https://en.wikipedia.org/wiki/Set_(mathematics)
Set difference.

RJB https://en.wikipedia.org/wiki/Set_(mathematics)
Complement of the set.

The symmetric difference of two sets ⁠X and Y⁠⁠, denoted ⁠ X Δ Y⁠, is the set of those elements that belong to X or Y but not to both: \[ ⁠X \Delta Y = (X \setminus Y) \cup (Y \setminus X) = \left\{ x \ : \ x \in X \vee x\in Y , \ x \notin X \cap Y \right\} . \]
RJB https://en.wikipedia.org/wiki/Set_(mathematics)
Symmetric difference of twosets.

Multisets

A multiset is a modification of the concept of a set that, unlike a set,[1] allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that element in the multiset. As a consequence, an infinite number of multisets exist that contain only elements a and b, but vary in the multiplicities of their elements.

 

 

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