Mathematics | Introduction of Set theory
Set is an unordered collection of objects, known as elements or members of the set. An element ‘a’ belong to a set A can be written as ‘a ∈ A’, ‘a ∉ A’ denotes that a is not an element of the set A.
Objects: Objects constituting a set are called its elements and we express the fact that ais an element of a set S, symbolically as a ∈ S.(Pronounce as a belongs to S).,
Since “object” is very general in scope, an element of a set may be letters, numbers or any symbole ordered pairs, or even a set itself. Sets are denoted by capita letters A, B,C etc and its element by small letters a,b,c …Sets are either be written by enumerating at its elements(listing method) or by a rule (rule method or set builder method)or by statement method.
(a): There are no particular order in a set, since it is only a collection of elements i.e. A = [1,2,3]
and B= [2,3,1] are both the same set. we say A= B.
(b): Repetition of element in a set, since an element is taken only once i.e. A= {1,2,2,3} and B = {1,2,3} are both the same sets.Representation of a Set A set can be represented by various methods. 3 common methods used for representing set: 1. Statement form. 2. Roaster form or tabular form method. 3. Set Builder method.
Statement form In this representation, a well-defined description of the elements of the set is given. Below are some examples of the same. 1. The set of all even number less than 10. 2. The set of the number less than 10 and more than 1. Roster form In this representation, elements are listed within the pair of brackets {} and are separated by commas. Below are two examples. 1. Let N is the set of natural numbers less than 5. N = { 1 , 2 , 3, 4 }. 2. The set of all vowels in the English alphabet. V = { a , e , i , o , u }. Set builder form In Set-builder set is described by a property that its member must satisfy. 1. {x : x is even number divisible by 6 and less than 100}. 2. {x : x is natural number less than 10}. Equal sets Two sets are said to be equal if both have same elements. For example A = {1, 3, 9, 7} and B = {3, 1, 7, 9} are equal sets.
NOTE: Order of elements of a set doesn’t matter. Subset A set A is said to be subset of another set B if and only if every element of set A is also a part of other set B. Denoted by ‘⊆‘. ‘A ⊆ B ‘ denotes A is a subset of B. To prove A is the subset of B, we need to simply show that if x belongs to A then x also belongs to B. To prove A is not a subset of B, we need to find out one element which is part of set A but not belong to set B.
‘U’ denotes the universal set. Above Venn Diagram shows that A is a subset of B. Size of a Set Size of a set can be finite or infinite. For example
Finite set: Set of natural numbers less than 100. Infinite set: Set of real numbers.
Size of the set S is known as Cardinality number, denoted as |S|. Example: Let A be a set of odd positive integers less than 10. Solution : A = {1,3,5,7,9}, Cardinality of the set is 5, i.e.,|A| = 5. Note: Cardinality of a null set is 0. Power Sets The power set is the set all possible subset of the set S. Denoted by P(S). Example: What is the power set of {0,1,2}? Solution: All possible subsets {∅}, {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, {0,1,2}. Note: Empty set and set itself is also the member of this set of subsets. Cardinality of power set is 
, where n is the number of elements in a set. Cartesian Products Let A and B be two sets. Cartesian product of A and B is denoted by A × B, is the set of all ordered pairs (a,b), where a belong to A and b belong to B.
A × B = {(a, b) | a ∈ A ∧ b ∈ B}.Example 1. What is Cartesian product of A = {1,2} and B = {p, q, r}. Solution : A × B ={(1, p), (1, q), (1, r), (2, p), (2, q), (2, r) }; The cardinality of A × B is N*M, where N is the Cardinality of A and M is the cardinality of B. Note: A × B is not the same as B × A.
Below are some Gate Previous question https://www.geeksforgeeks.org/gate-gate-cs-2015-set-2-question-28/ https://www.geeksforgeeks.org/gate-gate-cs-2015-set-1-question-26/ Set Theory continue
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