MATH’S January 2015 By: Ravi S.S.Sonu Set theory:- set is a collection of well defined and distinct objects. Representation of set :- Set-builder form Set-tabular or Roster form In roster form all the distinct objects are enclosed within braces { } and are separated by comma. In set-builder form all the elements of the set are defined by a common property. Consider a set of all the vowels of English alphabet. It is represented as follows:- Set-builder form:- A= {x I x is a vowel of English alphabet} or {x : x is a vowel of English alphabet} Roster form:- A= {a, e, i, o, u} or {e, a, i, u, o} (Here order of element does not matter) Symbol “∈” stands for element of a set and “∉” stands not an element of a set. Types of set:- Null/ Empty/ Void set:- A set having no element is called null/empty/void set. Ex.- A={x : x is an odd number divisible 2} Finite set:- A set in which number of element are countable is called finite set. Ex.- A={x : x is months of the year} Infinite set:- A set in which number of element are uncountable is called infinite set. Ex.- A={x : x is natural numbers} Singleton set or unit set or singlet:- A set in which only one element is present is called singleton set or unit set or singlet. It is a finite set. Ex.- A={x : x is an even prime number}. Pair set:- A set having only two elements is called pair set. It is also a finite set. Ex.- A={x : x is solution of x2=a} Subset:- If each and every element of set A is also an element of set B then A is called a subset of B. it is denoted by A ⊆ B and is read as A is subset of B. Ex.- if A={1, 2, 4}; B={1, 2, 3, 4} here all element of set A is in set B so A is subset of B and written as A⊆ B. The null set or empty set ϕ is a subset of every set. If a set has n elements then it has 2n subsets. If A is a subset of B and B is a subset of C then A is also an element of set B. i.e.-A⊆B and B⊆C then A⊆C. Equal sets:- Two sets A and B are said to be equal if each element of set A is also an element of set B. If A and B are two equal sets then by definition of subset A is subset of B and B is subset of A then A⊆B and B⊆A ⇔ A=B i.e. A⊆B and B⊆A then A=B and if A=B then A⊆B and B⊆A. Proper subset:- set A is called a proper subset of set B if it follows the conditions :- Every element of set A must be in the set B. There is at least one element in set B, which is not in set A. Ex.- if A={1, 2, 3} then subsets are { }, {1},{2},{3},{1, 2},{1, 3},{2,3},{1, 2, 3}. B={1, 2, 3, 4} then A is a proper subset of set B and is denoted as A⊂B. NOTE:- Every proper subset is a subset but every subset is not a proper subset. Super set:- If A is a subset of B then B is called superset of A. It is denoted by B⊇A. Ex.- if A={1, 2, 3} and B={1, 2, 3, 4} then A⊆B, i.e. A is subset of B or it is also written as B⊇A, i.e. B is superset of A. Power set:- The set of all the subsets of a given set A is called the power set of the set A. it is denoted by P(A). Ex.- If A={1, 2, 3} then its subsets are {},{1},{2},{3},{1, 2},{1, 3},{2, 3},{1, 2, 3} , its power set P(A)={ {},{1},{2},{3},{1, 2},{1, 3},{2, 3},{1, 2, 3 }} Symbolically:- P(A)= {B:B⊆A} ***If a set A has n elements then its power set P(A) has 2n elements.*** Universal set:- If all the sets under consideration are subsets of a given set then the given set is called the universal set. It is denoted by S or U. Ex.: if A={1, 2, 3}, B={2, 3, 4, 5} and C ={5, 6, 7, 8} are the three sets Now let us consider S or U ={1, 2, 3, 4, 5, 6, 7, 8, 9} here A,B and C are subset of set S or U so S or U can be consider as universal set. Equivalent sets:- If number of elements of two sets are equal then they are called equivalent sets. Ex.- if A={1, 2, 3} and B={4, 5, 6} then A and B are called equivalent sets, because no of element in A = no of element in B. Disjoint sets:- Two sets A and B are said to be disjoint, if they have no common element. Ex.- if A={1, 2} and B={3, 4, 5} then A and B are disjoint sets. Interval notations:-Inequality Notation Interval Notation Number Line Notation 1). 1≤X≤3 X∈ [1, 3] 3 2). 1
Posted on: Sat, 03 Jan 2015 17:43:50 +0000
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