Types of sets

Meaning and types of sets.

1. Types of Sets Mrs.Mahitha Davala M.Com ., M.B.A .

2. A set is a collection of well defined objects. The objects are well distinguished also. e g ., a set of all natural numbers from1 to 50. The numbers are ( i ) well defined (ii) Well distinguished The objects that constitutes set are called its members or elements.

3. Description of Set A set is described in the following two ways: ( i )Roster method : Under this method a set is described by listing elements, seperated by commas, within braces{ }. Eg ., The set of vowels of English Alphabet may be described as { a,e,i,o,u }. The order in which the elements are written in a set does not make any difference. Therefore, { a,e,i,o,u } and { e,a,i,o,u } denote the same set.

4. (ii) Set-Builder method :Under this method , a set is described by a characterizing property P(x) of its elements x. In such a case the set is described by {x:P(x) holds} or, { x│P (x) hols }, which is read as “the set of all x such that P(x) holds”. The symbol ‘│, or ‘:’ is read as’such that’. Eg ., The set A={1,2,3,4,5,6,7} can be written as A={x ∈N :x≤7}

5. Types of Sets Empty Set : A set is said to be empty or null or void set if it has no element and it is denoted by ∅. A set consisting of atleast one element is called a non empty or non-void set. Singleton set : A set consisting of a single element is called a singleton set . Eg ., the set{6} is a singleton set.

6. Finite Set : A set is called a finite set it its is either void set or its elements can be listed, counted,labelled by natural number ‘n’ (say). Cardinal Number of a Finite Set: The number ‘n’ in the above definition is called the cardinal number or order of a finite set A and is denoted by n(A). Infinite Set : A set whose elements cannot be listed by the natural numbers 1,2,3,….n for any natural number n is called an infinite set.

7. Equivalent Set : Two finite sets A and B are equivalent if their cardinal numbers are same i.e., n(A) = n(B). Equal Sets : Two sets A and B are said to be equal if every element of A is a member of B and every element of B is a member of A. If sets are equal we write A=B and A≠B , when A and B are not equal.

8. Subsets : Let A and B be two sets. If every element of A is an element of B, then A is called a subset of B. If A is a subset of B, we write A   ⊆ B, which is read as “ A is a subset of B” or “A is contained in B”. Thus, A  ⊆ B if a ∈A ⇒ a ∈ B. If A is a subset of B, we say that B contains A or B is a Super set of A and we write B ⊃A. If A is not a subset of B, we write A ⊄ B. Every set is a subset of itself and the empty set is subset of every set. These two subsets are called improper subsets.

9. Universal Set : In set theory, there is a set that contains all sets under consideration i.e., it is a super set of each of the given sets. Such a set is called the Universal set and is denoted by U. In other words, a set that contains all sets in a given context is called the universal set. If A={1,2,3,4}, B={2,3,4,5} and C= {1,3,5,7}, the U={1,2,3,4,5,6,7} can be taken as the universal set.

10. Power set : Let A be a set. Then the collection of family of all subsets of A is called the power set of A and is denoted by P(A). Let A={1,2,3}. Then the subsets of A are: ∅,{1},{2},{3},{1,2},{1,3},{2,3} and {1,2,3}. Hence P (A)= {∅, ,{1},{2},{3},{1,2},{1,3},{2,3} {1,2,3 }}. The number of proper subsets in a finite set is obtained by n(⊂)=2ⁿ -1 where, n represents the number of elements in a finite set and n(⊂ )represents the number of proper subsets of the said set.

11. Thank You