Applied Mathematics: Operation on Sets and Venn Diagrams
Set
Set is a collection of well-defined objects which are distinct from each other. The objects in the set are called its elements. Sets are usually denoted by capital letters A, B, C, ....... and elements are usually denoted by small letters a, b, c, ........ For example, the set of all even natural numbers less than 10 can be represented by N = {2, 4, 6, 8}.
Methods for describing a set
(i) Roster Method: In this method, a set is described by listing elements, separated by commas, within braces, e.g. A = {a, e, i, o, u}
Note: This method is also called listing method or tabular form method.
(ii) Set builder method: In this method, we write down a rule which gives us all the elements of the set by that rule e.g. A = {x : x is a vowel of English alphabet}
Finite Set: A set containing finite number of elements or no element, is called a finite set.
e.g. The set of all persons in India is a finite set.
Infinite Set: A set containing infinite number of elements is called an infinite set.
Cardinality of a Finite Set: The number of elements in a given finite set is called cardinal number of finite set, denote by n (a), where A is the given set.
e.g. P = {5, 15, 25, 35, 45} \[\Rightarrow \] n (P) = 5
Empty Set (or null set)
A set containing no element in it, is called on empty set. It is represented by {} or \[\phi \] (read as phi), e.g. The set of all whole numbers less than o.
Singleton Set
A set containing a single element is called a singleton set. e.g. The set of ail prime numbers.
Equal Sets: Two sets A and B are called equal, if every element of A is a member of B and every element of B is a member of A. Thus we write A = B.
e.g. A = {2, 4, 6, 8, 10,} and {all the even natural numbers less than or equal to 10} ie. A and B are equal sets.
Find cardinal number of a set A of the composite numbers between 10 and 25.
(a) 4 (b) 6
(c) 8 (d) 9
(e) None of these
Answer (d)
Explanation: Here, A = {12, 14, 15, 16, 18, 20, 21, 22, 24}
\[\Rightarrow n\,(A)\,=\,9\,\Rightarrow \]Cardinal number of set A = 9
Disjoint Sets
Two sets A and B are called to be disjoint, if they have no elements in common. e.g. Sets A {2, 4, 6, 8, 10, 12} and B = {1, 3, 5, 7, 9, 11} \[\Rightarrow A\cap B\,\,=\phi \] \[\Rightarrow \]A
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