A set must be well defined; i.e., for any given object, it must be unambiguous whether or not the object is an element of the set. For example, if a set contains all the chairs in a designated room, then any chair can be determined either to be in or not in the set. If there were no chairs in the room, the set would be called the empty, or null, set, i.e., one containing no elements. A set is usually designated by a capital letter. If A is the set of even numbers between 1 and 9, then A={2, 4, 6, 8}. The braces, {}, are commonly used to enclose the listed elements of a set. The elements of a set may be described without actually being listed. If B is the set of real numbers that are solutions of the equation x2=9, then the set can be written as B={x:x2=9} or B={x|x2=9}, both of which are read: B is the set of all x such that x2=9; hence B is the set {3,-3}.
Membership in a set is indicated by the symbol ∈ and nonmembership by ∉; thus, x∈A means that element x is a member of the set A (read simply as “x is a member of A”) and y∉A means y is not a member of A. The symbols ⊂ and ⊃ are used to indicate that one set A is contained within or contains another set B; A⊂B means that A is contained within, or is a subset of, B; and A⊃B means that A contains, or is a superset of, B.SET, in mathematics, collection of entities, called elements of the set, that may be real objects or conceptual entities. Set theory not only is involved in many areas of mathematics but has important applications in other fields as well, e.g., computer technology and atomic and nuclear physics.
A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.
Types Of Sets
NULL/ VOID/ EMPTY SET
A set which has no element is called the null set or empty set andis denoted by Φ(phi). The number of elements of a set A is denoted as n (A) and n (Φ) = 0 as it contains no element. For example the set of all real numbers whose square is –1.SINGLETON SET
A set containing only one element is called Singleton Set.FINITE AND INFINITE SET
A set, which has finite numbers of elements, is called a finite set. Otherwise it is called an in finite set. For example, the set of all days in a week is a finite set whereas; the set of all integers is an infinite set.UNION OF SETS
Union of two or more sets is the set of all elements that belong to any of these sets. The symbol used for union of sets is ‘∪’ i.e. A∪B = Union of set A and set B = {x: x A or x B (or both)}Example: A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8}, then A∪B∪C = {1, 2, 3, 4, 5, 6, 8}
INTERSECTION OF SETS
It is the set of all the elements, which are common to all the sets. The symbol used for intersection of sets is ‘∩’ i.e. A ∩ B = {x: x A and x B}
Example:If A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8}, then A ∩ B ∩ C = {2}
DIFFERENCE OF SETS
The difference of set A to B denoted as A – B is the set of those elements that are in the set A but not in the set B i.e. A – B = {x: x A and x ∉ B}Similarly B – A = {x: x B and x ∉ A}
In general A–B ¹ B–A
Example: If A = {a, b, c, d} and B = {b, c, e, f} then A–B = {a, d} and B–A = {e, f}.
Symmetric Difference of Two Sets:
For two sets A and B, symmetric difference of A and B is given by (A – B) ∪ (B – A) and is denoted by A B.
SUBSET OF A SET
A set A is said to be a subset of the set B if each element of the set A is also the element of the set B. The symbol used is ‘⊆’ i.e. A ⊆ B (x A => x B).Each set is a subset of its own set. Also a void set is a subset of any set. If there is at least one element in B which does not belong to the set A, then A is a proper subset of set B and is denoted as A ⊂ B. e.g If A = {a, b, c, d} and B = {b, c, d}. Then B ⊂ A or equivalently A ⊃ B (i.e A is a super set of B). Total number of subsets of a finite set containing n elements is 2n.
Equality of Two Sets:
Sets A and B are said to be equal if A ⊆ B and B ⊆ A; we write A = B.
DISJOINT SETS
If two sets A and B have no common elements i.e. if no element of A is in B and no element of B is in A, then A and B are said to be Disjoint Sets. Hence for Disjoint Sets A and B => n (A ∩ B) = 0.
OPERATION OF SETS:
Unions
The union of A and B, denoted A ∪ B
Main article: Union (set theory)
Two sets can be "added" together. The union of A and B, denoted by A ∪ B, is the set of all things which are members of either A or B.Examples:
· {1, 2} ∪ {red, white} ={1, 2, red, white}.
· {1, 2, green} ∪ {red, white, green} ={1, 2, red, white, green}.
· {1, 2} ∪ {1, 2} = {1, 2}.
Some basic properties of unions:· A ∪ B = B ∪ A.
· A ∪ (B ∪ C) = (A ∪ B) ∪ C.
· A ⊆ (A ∪ B).
· A ∪ A = A.
· A ∪ ∅ = A.
[edit] Intersections
Main article: Intersection (set theory)
A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A ∩ B, is the set of all things which are members of both A and B. If A ∩ B = ∅, then A and B are said to be disjoint.
The intersection of A and B, denoted A ∩ B.
Examples:· {1, 2} ∩ {red, white} = ∅.
· {1, 2, green} ∩ {red, white, green} = {green}.
· {1, 2} ∩ {1, 2} = {1, 2}.
Some basic properties of intersections:· A ∩ B = B ∩ A.
· A ∩ (B ∩ C) = (A ∩ B) ∩ C.
· A ∩ B ⊆ A.
· A ∩ A = A.
· A ∩ ∅ = ∅.
[edit] Complements
The relative complement
of B in A
of B in A
The complement of A in U
The symmetric difference of A and B
Main article: Complement (set theory)
Two sets can also be "subtracted". The relative complement of B in A (also called the set-theoretic difference of A and B), denoted by A \ B (or A − B), is the set of all elements which are members of A but not members of B. Note that it is valid to "subtract" members of a set that are not in the set, such as removing the element green from the set {1, 2, 3}; doing so has no effect.In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply complement of A, and is denoted by A′.
Examples:
· {1, 2} \ {red, white} = {1, 2}.
· {1, 2, green} \ {red, white, green} = {1, 2}.
· {1, 2} \ {1, 2} = ∅.
· {1, 2, 3, 4} \ {1, 3} = {2, 4}.
· If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then E′ = O.
Some basic properties of complements:· A \ B ≠ B \ A.
· A ∪ A′ = U.
· A ∩ A′ = ∅.
· (A′)′ = A.
· A \ A = ∅.
· U′ = ∅ and ∅′ = U.
· A \ B = A ∩ B′.
An extension of the complement is the symmetric difference, defined for sets A, B as
[edit] Cartesian product
Main article: Cartesian product
A new set can be constructed by associating every element of one set with every element of another set. The Cartesian product of two sets A and B, denoted by A × B is the set of all ordered pairs (a, b) such that a is a member of A and b is a member of B.Examples:
· {1, 2} × {red, white} = {(1, red), (1, white), (2, red), (2, white)}.
· {1, 2, green} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green), (green, red), (green, white), (green, green)}.
· {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
Some basic properties of cartesian products:· A × ∅ = ∅.
· A × (B ∪ C) = (A × B) ∪ (A × C).
· (A ∪ B) × C = (A × C) ∪ (B × C).
Let A and B be finite sets. Then· | A × B | = | B × A | = | A | × | B |.
