Title 1. SETS.pdf ✅

Mathematics A set which is not a finite set is called an infinite set. Thus a set A is said to be an infinite set if the number of elements of set A is not finite. Example:Let A = set of all points on a particular straight line. 1.3 CARDINAL NUMBER OF A FINITE SET The number of elements in a finite set A is called the cardinal number of set A and is denoted by n(A) Example:Let A = {1, 2, 3, 4, 5}, then n(A) = 5 1.4 EQUIVALENT SETS Two finite sets A and B are said to be equivalent if they have the same cardinal number. Thus set A and B are equivalent iff n(A) = n(B). If sets A and B are equivalent, we write A  B Example: Let A = {1, 2, 3, 4, 5}, B = {a, e, i, o, u} Here n(A) = n(B) = 5 Therefore, sets A and B are equivalent. 1.5 EQUAL SETS Two set A and B are said to be equal set if each element of set A is an element of set B and each element of B is an element of set A. Thus two sets A and B are equal if they have exactly the same elements. The order in which the elements in the two sets have been written is immaterial. If set A and B are equal we can write A = B Example1: Let A = {1, 2, 3, 4, 5}, B = { x : x N and 1 x  5 } Here A and B are equal. 2.1 NULL SET (OR EMPTY SET OR VOID A SET) A set having no element is called null set or empty set or void set. It is denoted by  or {}. Example:The set of odd numbers divisible by 2. 2.2 SINGLETON SET A set having single element is called a singleton set. It is represented by writing down the element within the braces. Example:{2}, {0}, {}. 2.3 UNIVERSAL SET A set consisting of all possible elements which occur in the discussion is called a universal set and is denoted by U. 2.4 PAIR SET A set having two elements is called a pair set. Example:{1, 2}, {2, 0}. 2.5 SET OF SETS A set S having all its elements as set is called a set of sets or a family of sets or a class of sets. Example1: S = 1, 2, 3, 3, 4 is not a set of sets as 3 is not a set. Example2:{} is a singleton set of set having null set  as its elements. DIFFERENT TYPES OF SETS 2
Mathematics 3.1 SUBSETS OF A SET A set A is said to be a subset of a set B if each element of A is also an element of B. If A is a subset of set B, we write A  B Thus, A  B  [x  A  x B] Example:Let A = {1, 2, 3}, B = {2, 3, 4, 1, 5}, then A  B . The statement A  B can also be expressed equivalently by writing B  A (read ‘B is a superset of A’ ) If A is not a subset of B i.e., if there is an element in A which is not an element of B, then we write A  B or B  A. • Some important properties of subset • Every set is its own subset. Let A be any set ; x  A  x  A Hence A  A • Empty set is a subset of each set. • Let A and B be any two sets: then A = B  A  B and B  A • Let A, B, C be three sets. If A  B and B  C , then A  C . 3.2 PROPER SUBSET OF A SET A set A is said to be a proper subset of a set B, if A is a subset of B and A  B i.e. if Every element of A is an elements of B and B has at least one element which is not an element of A. This fact is expressed by writing A  B or B  A. If A is not a proper subset of B, then we write A  B . Example: Let A = 1, 2, 3 and B = 2, 3, 4, 1, 5 , then A  B and B  A. 3.3 SUPERSET OF SETS A set A is said to be a super set of set B, if B is a subset of A i.e., each elements of B is an elements of A. If A is a super set of B, then A  B . Example:Let A = {1, 2, 3, 4, 5) and B = {2, 5, 4}. Here B is a subset of A, therefore A is a superset of B. 3.4 POWER SET The set or family of all the subsets of a given set A is said to be the power set of A and is denoted by P(A) Example: If A = {1, 2} P(A) = , 1, 2, 1, 2 If A has n elements then P(A) has 2nelements. Illustration 1 Question: List all the subsets and all the proper subsets of the set {–1, 0, 1}. Solution: Let A = {−1, 0, 1} . Subset of A having no element is :  Subsets of A having one element are : {–1}, {0}, {1}. Subsets of A having two elements are : {–1, 0}, {0, 1}, {–1, 1}. Subsets of A having three elements are : {–1, 0, 1}. Thus, all the subsets of A are , {–1}, {0}, {1}, {–1, 0}, {0, 1}, {–1, 1}, {–1, 0, 1}. Proper subsets of A are , {–1}, {0}, {1}, {–1, 0}, {0, 1}, {–1, 1}. SUBSETS, SUPERSETS, PROPER SUBSETS 3
Mathematics Illustration 2 Question: Make correct statements by filling the blanks by suitable symbols / ,  . (i) {x : x is an even natural number} –––––– {x : x is an integer} (ii) {x : x is a triangle in the plane} –––––– {x : x is a rectangle in the plane} (iii) {x : x is isosceles triangle in the plane} –––––– {x : x is an equilateral triangle in the plane} (iv) a–––––– {a, {b}, c} (v) {{ }} { , { }, } a ______ a b c Solution: (i) Since every even natural number is an integer, therefore, {x : x is an even natural number}  {x : x is an integer}. (ii) Since a triangle is not a rectangle, therefore {x : x is a triangle in the plane } / {x : x is a rectangle in the plane}. (iii) Since an isosceles triangle is not necessarily an equilateral triangle, therefore { x : x is an isosceles triangle} / {x : x is an equilateral triangle}. (iv) Since a is not a set, therefore, a / {a, {b}, c}. (v) Since {{a}} is a set containing exactly one element {a} and {a} is not an element of the set {a, {b}, c}, therefore, {{a}} / {a, {b}, c}. Illustration 3 Question: How many elements are in the set A = {, {}, {, {}}} B = {x : x is even integer and x< 19} C = {x : 0 x  1 and x is a rational number} Solution: The elements of A are , {}, {, {}}. So A has three elements. B = {x : x = 0,  2,  4,  6, ... and x  19} = {..., –4, –2, 0, 2, 4, 6, ..., 18}  B is an infinite set. C is also infinite set because , .... 4 1 , 3 1 , 2 1 1, are all elements of C. Important formulae/points • The order in which the elements of a set are written is immaterial thus the set {1, 2,3} and {2, 1, 3} are same. • Two sets A and B are equal if x  A  x B and x B  x  A. • The set {0} is not an empty set as it contains one element 0. • The set {} is not an empty set as it contain one element . • A  B  P(A) P(B) • If A has n elements then P(A) has n 2 elements.