A set means a well-defined collection of objects. Set is denoted by A, B, C, X, Y, Z, ... etc.
The objects in a set are called elements of the sets are denoted by a, b, c, x, y, z, etc.
If x is a member of set A, then we write x∈A, which is read as x belongs to A.
Types of sets
A set consisting of only one elements is called a singleton set.
A set which does not contain any element is called an empty set. A set which is not empty is called a non-empty set.
Generally, when we consider many sets of similar nature the elements in the sets are selected from a definite set. This set is called the universal and it is denoted by U.
A set A is said to be subset of a set B if every element of A is also an element set B. If a set A is a subset of a set B then B is called super set of A.
For any set A, the set consisting of all the subsets of A is called the power set of A and it is denoted by P(A).
Two sets A and B are said to be equal sets, if they have the same elements. Thus if for all x if x∈A, then x∈B and if for all x, if x∈B then x∈A, then A = B.
Finite and infinite sets
Finite set - A set is said to be finite if it has finite number of elements.
Infinite set - A set is said to be infinite if it has an infinite number of elements.
Operations on sets
Union of sets
Let A, B ∈ P(U); The set consisting of all elements of U which are in A or in B is called the union of sets A and B and it is denoted by A ∪ B. The operation of taking the union of two sets is called the union operation.
Intersection of sets
Let A, B ∈ P(U); Then the set consisting of all elements of U which are in both A and B is called the intersection set of sets A and B and is denoted by A ∩ B. The operation of finding the intersection of two sets is called the intersection operation.
Union of Sets
Intersection of Sets
Non-empty sets A and B are said to be disjoint if their intersection is the empty set. If A and B are disjoint sets, then A ∩ B = Φ.
For A∈P(U), the set consisting of all those elements of U which are not in A, is called complement of A and is denoted by A'.
For the sets A, B ∈ P(U), the set consisting of all elements of A which are not in B, is called the difference set A and B, This set is denoted by A - B. The operation of taking the difference of two sets is called the difference operation.
Symmetric Difference Set
For sets, A, B ∈ P(U), the set consisting of all elements which are in the set A or in the set B, but not in both is called symmetric difference of the set A and B. Symmetric difference of two sets is denoted by A Δ B.
Cartesian Product of Sets
Let A and B be two non-empty sets. Then, the set of all ordered pairs (x, y), where x ∈ A, y ∈ B is called cartesian product of A and B. It is denoted by A X B (read as A cross B).
Number of Elements of a Finite Set
n(A) denotes the number of elements in a finite set A.