Selina Publishers Concise Maths Class 7 ICSE Solutions\u00a0Chapter 13 Set Concepts (Some Simple Divisions by Vedic Method)<\/strong><\/p>\n ICSE Solutions<\/a>Selina ICSE Solutions<\/a>ML Aggarwal Solutions<\/a><\/p>\n APlusTopper.com provides step by step solutions for Selina Concise ICSE Solutions for Class 7 Mathematics. You can download the Selina Concise Mathematics ICSE Solutions for Class 7 with Free PDF download option. Selina Publishers Concise Mathematics for Class 7 ICSE Solutions all questions are solved and explained by expert mathematic teachers as per ICSE board guidelines.<\/p>\n Selina Class 7 Maths ICSE Solutions<\/a>Physics<\/a>Chemistry<\/a>Biology<\/a>Geography<\/a>History & Civics<\/a><\/p>\n POINTS TO REMEMBER<\/strong> Question 1.<\/strong><\/span> Question 2.<\/strong><\/span> Question 3.<\/strong><\/span> Question 4.<\/strong><\/span> Question 5.<\/strong><\/span> Question 6.<\/strong><\/span> Question 1.<\/strong><\/span> Question 2.<\/strong><\/span> Question 3.<\/strong><\/span> Question 4.<\/strong><\/span>
\n1. Definition of a Set :<\/strong> In our day to day life, different collective nouns are used to describe collection of objects ; such as : a group of students playing cricket, a pack of cards, a bunch of flowers, etc
\nIn mathematics, such collections of objects are named as sets.
\nA set is a collection of well-defined objects, things or symbols, etc.<\/strong>
\nThe phrase \u2018well-defined\u2019 means ; it must be possible to know, without any doubt, whether a given object (thing or symbol) belongs to the set under consideration or not.
\nFor example :<\/strong>
\n\u201cThe set of tall boys of Class 10\u201d is not well-defined ; since it is not possible to know that which boys are to be included and exactly what is the limit.
\nBut when we say, \u201cThe set of boys of Class 10, which are taller than Peter\u201d, now we can compare the heights of different boys with the height of Peter and can know exactly, that which boys are to be included in the required set. Thus, the objects are well-defined.
\n2. Elements of a set:<\/strong> The objects (things, symbols, etc.) used to form a set are called elements or members of the set.
\nIn general, a set is denoted by a capital letter of English alphabet with its elements written inside curly braces and separated by commas.
\ne.g., Set A = {5, 10, 12, 15}
\n3. Use of Symbol \u2018\u2208\u2019 or Symbol \u2018\u2209 \u2018 :<\/strong> The symbol \u2018\u2208\u2019 stands for \u2018belongs to\u2019 or \u2018 is an element of\u2019 or \u2018is a member of\u2019; whereas the symbol \u2018\u2209\u2019 stands for \u2018does not belong to\u2019 or \u2018is not an element of or \u2018is not a member of\u2019.
\ne.g., For set P = {3, 6, 8, 13, 18} ; 3 \u2208 P, 5\u2209 P and so on.
\n(i)<\/strong> The elements in a set can be written in any order.
\nThus, {a, b, c, d] is the same set as {b, d, a, c} or {c, b, d, a}, etc.
\n(ii)<\/strong> The elements in a set should not be repeated, i.e. if any element occurs many times, it should be written only once.
\nThus, set of letters of the word \u2018crook\u2019 = {c, r, o, k}.
\nThere are two os in the given word \u201ccrook\u201d ; but in the set, it is written only once.
\n4. Representation of A Set:<\/strong> A set, in general, is represented in :
\n(i) Description method (form)
\n(ii) Tabular or Roster method (form)
\n(iii) Set-builder or Rule method
\nFor example :<\/strong>
\nN is the set of natural numbers [Description method]
\nN = {1, 2, 3, 4, 5, ……. } [Roster or Tabular method]
\nN = {x : x is a natural number}, or {x : x \u2208 N} [Set-builder or Rule method]
\n[The symbol \u2018 : \u2019 stands for such that and the set {x : x \u2208 N} is read as, \u201cthe set of x such that x is a natural number\u201d].
\nIt is clear from the example given above that:
\n(i)<\/strong> in description method a well-defined description about the set is given.
\n(ii)<\/strong> in roster or tabular method the elements of the set are written inside a pair of curly braces and are separated by commas.
\n(iii)<\/strong> in set-builder or ruler method the actual elements of the set are not written, but a rule or a statement or a formula is written in the briefest possible way.
\n5. Cardinal Number :<\/strong> The cardinal number of a set is the number of elements in it.
\nThus, if a set A has 5 elements ; its cardinal number is 5 and we represent it by writing n (A) = 5 .
\nSimilarly, if set B = Set of even natural numbers less than 10 then, B = {2, 4, 6, 8} and n (B) = 4.
\nIf B = {0}, then n (B) = 1. Since, 0 is an element of set B.<\/strong>
\n6. Types of Sets :<\/strong>
\n(i) Finite Set:<\/strong> A set is said to be a finite set, if it has a limited (countable) number of elements in it.
\nFor example :<\/strong>
\n(a) S = Set of natural numbers between 10 and 15 = {11, 12, 13, 14}
\n(b) P = {0, 1,2, , 20} = {x : x \u2208 W and x \u2264 20} and so on.
\n(ii) Infinite set:<\/strong> A set is said to be an infinite set, if it has an unlimited (uncountable) number of elements in it.
\nFor example :<\/strong>
\n(a) P = Set of prime numbers = {2, 3, 5,…. }
\n(b) B = (x : x \u2208 N and x \u2265 21} = (21, 22, 23, } ans so on.
\n(iii)<\/strong> Empty set or Null set:<\/strong> The set, with no element in it, is called the empty set or the null set.
\nThe empty set is represented by a pair of braces with no element in it or by the Danish letter \u03a6, which is pronounced as \u2018oe’
\nThus, the empty set = { } = \u03a6
\nNote :<\/strong> For empty set, it is wrong to call \u2018an empty set<\/strong>\u2019 or \u2018a null set<\/strong>\u2019 as there is one and only one empty set though it may have many descriptions.
\nTherefore, it is always called \u201cthe empty set or the null set\u201d.<\/strong>
\nSome examples of the empty set :<\/strong>
\n(a) Let A = {a man of age more than 400 years}.
\nSince there can not be any man with the age more than 400 years; the set A will have no element in it i.e. It is the empty set. And we write : A = { } or \u03a6
\n(b) If B = (Triangles with 4 sides} ; it is clear that B = \u03a6
\nNote :<\/strong>
\n1. \u03a6 \u2260{0}, since {0} is a set with 0 as its element whereas \u03a6 has no element.
\n2. {\u03a6} \u2260 {0}, since both the sets have different elements.
\n3. The cardinal number of the empty set is 0 i.e. n (\u03a6) = 0.
\n(iv) Disjoint sets : Sets having no element in common are called disjoint sets.
\nFor example :<\/strong>
\nSets P = (5, 7, 9} and O = (4, 6, 10, 12} are disjoint; as they do not have any element in common.
\n(v) Joint (overlapping) sets :<\/strong> Sets having atleast one element in common are called joint or overlapping sets.
\nFor example:<\/strong>
\nSet B = (4,6, 8,10,12} and set C = (3,6,9,12,15} are joint sets; as they have elements 6 and 12 common.
\n(vi) Equal sets :<\/strong> Two sets are said to be equal, if the elements of both the sets are the same.
\nFor example :<\/strong>
\nIf set A = {x, y, z} and set B = {last three letters of English alphabet}.
\nClearly, sets A and B have the same elements and so set A = set B.<\/strong>
\n(vii) Equivalent sets :<\/strong> Two sets are said to be equivalent, if they have equal number of elements in them, i.e. the cardinal numbers of both the sets are equal.
\nFor example :<\/strong>
\nLet A = (3, 6, 9} and B = {a, b, c}.
\nSince, set A has 3 elements and set B also has 3 elements i.e., n (A) = n (B); therefore, sets A and B are equivalent and for this, we write : A \u2194 B.
\nNote:<\/strong>
\n1. Equal sets are always equivalent; but the converse is not always true (i.e. it is not necessary that equivalent sets are equal also).
\n2. In equivalent sets, the number of elements (cardinal number) are equal, whereas in equal sets the element are the same.
\n3. Two infinite sets are always equivalent;<\/p>\nSet Concepts Exercise 13A –\u00a0Selina Concise Mathematics Class 7 ICSE Solutions<\/h3>\n
\nFind, whether or not, each of the following collections represent a set:<\/strong>
\n (i) The collection of good students in your school.<\/strong>
\n (ii) The collection of the numbers between 30 and 45.<\/strong>
\n (iii) The collection of fat-people in your colony.<\/strong>
\n (iv) The collection of interesting books in your school library.<\/strong>
\n (v) The collection of books in the library and are of your interest.<\/strong>
\nSolution:<\/strong><\/span>
\n(i)<\/strong> It is not a set as it is not well defined.
\n(ii)<\/strong> It is a set
\n(iii)<\/strong> It is not a set as it is not well defined.
\n(iv)<\/strong> It is not a set as it is not well defined.
\n(v)<\/strong> It is a set.<\/p>\n
\nState whether true or false :<\/strong>
\n (i) Set {4, 5, 8} is same as the set {5, 4, 8} and the set {8, 4, 5}<\/strong>
\n (ii) Sets {a, b, m, n} and {a, a, m, b, n, n) are same.<\/strong>
\n (iii) Set of letters in the word \u2018suchismita\u2019 is {s, u, c, h, i, m, t, a}<\/strong>
\n (iv) Set of letters in the word \u2018MAHMOOD\u2019 is {M, A, H, O, D}.<\/strong>
\nSolution:<\/strong><\/span>
\n(i)<\/strong> True
\n(ii)<\/strong> True
\n(iii)<\/strong> True as it has the same elements
\n(iv)<\/strong> True as it has the same elements.<\/p>\n
\nLet set A = {6, 8, 10, 12} and set B = {3, 9, 15, 18}.<\/strong>
\n Insert the symbol \u2018 \u2208 \u2019 or \u2018 \u2209 \u2019 to make each of the following true :<\/strong>
\n (i) 6 …. A<\/strong>
\n (ii) 10 …. B<\/strong>
\n (iii) 18 …. B<\/strong>
\n (iv) (6 + 3) …. B<\/strong>
\n (v) (15 – 9) …. B<\/strong>
\n (vi) 12 …. A<\/strong>
\n (vii) (6 + 8) …. A<\/strong>
\n (viii) 6 and 8 …. A<\/strong>
\nSolution:<\/strong><\/span>
\n(i)\u00a0<\/strong>6 \u2208 A
\n(ii)<\/strong> 10 \u2209 B
\n(iii)<\/strong> 18 \u2208B
\n(iv)<\/strong> (6 + 3) or 9 \u2208 B
\n(v)<\/strong> 15 – 9 or 6 g B
\n(vi)<\/strong> 12 \u2208 A
\n(vii)<\/strong> 6 + 8 or 14 \u2209 A
\n(viii)<\/strong> 6 and 8 \u2208 A<\/p>\n
\nExpress each of the following sets in<\/strong>
\n roster form :<\/strong>
\n (i) Set of odd whole numbers between 15 and 27.<\/strong>
\n (ii) A = Set of letters in the word \u201cCHITAMBARAM\u201d<\/strong>
\n (iii) B = {All even numbers from 15 to 26}<\/strong>
\n (iv) P = {x : x is a vowel used in the word \u2018ARITHMETIC\u2019}<\/strong>
\n (v) S = {Squares of first eight whole numbers}<\/strong>
\n (vi) Set of all integers between 7 and 94; which are divisible by 6.<\/strong>
\n (vii) C = {All composite numbers between 2 and 20}<\/strong>
\n (viii) D = Set of Prime numbers from 2 to 23.<\/strong>
\n (ix) E = Set of natural numbers below 30 which are divisible by 2 or 5.<\/strong>
\n (x) F = Set of factors of 24.<\/strong>
\n (xi) G = Set of names of three closed figures in Geometry.<\/strong>
\n (xii) H = {x : x eW and x < 10}<\/strong>
\n (xiii) J = {x: x e N and 2x – 3 \u226417}<\/strong>
\n (xiv) K = {x : x is an integer and – 3 < x < 5}<\/strong>
\nSolution:<\/strong><\/span>
\n(i)<\/strong> {17, 19, 21, 23, 25}
\n(ii)<\/strong> A = (C, H, I, T, A, M, B, R}
\n(iii)<\/strong> B = {16, 18, 20, 22, 24, 26}
\n(iv)<\/strong> P = {a, e, i}
\n(v)<\/strong> S = {0, 1, 4, 9, 16, 25, 36, 49}
\n(vi)<\/strong> {12, 18, 24; 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90}
\n(vii)<\/strong> C = {4, 6, 8, 9, 10, 12, 14, 15, 16, 18}
\n(viii)<\/strong> D = {2, 3, 5, 7, 11, 13, 17, 19,23}
\n(ix)<\/strong> E = {2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 28}
\n(x)<\/strong> F={l,2, 3, 4, 6, 8, 12, 24}
\n(xi)<\/strong> G = {Triangle, quadrilateral, circle}
\n(xii)<\/strong> H = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
\n(xiii)<\/strong> 2x – 3 \u2264 17
\n\u21d2 2x \u2264 17 + 3 2 x \u2264 20
\n\u21d2 x \u2264 \\(\\frac { 20 }{ 2 }\\)
\nx \u2264 10
\n\u2234 J = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
\n(xiv)<\/strong> \u2235 – 3 < x < 5
\n\u2234x lies between – 3 and 5
\n\u2234K = {- 2, – 1, 0, 1, 2, 3, 4}<\/p>\n
\nExpress each of the following sets in set- builder notation (form) :<\/strong>
\n (i) {3, 6, 9, 12, 15}<\/strong>
\n (ii) {2, 3, 5, 7, 11, 13 …. }<\/strong>
\n (iii) {1, 4, 9, 16, 25, 36}<\/strong>
\n (iv) {0, 2, 4, 6, 8, 10, 12, …. }<\/strong>
\n (v) {Monday, Tuesday, Wednesday}<\/strong>
\n (vi) {23, 25, 27, 29, … }<\/strong>
\n (vii) {\\(\\frac { 1 }{ 3 }\\),\\(\\frac { 1 }{ 4 }\\),\\(\\frac { 1 }{ 5 }\\),\\(\\frac { 1 }{ 6 }\\),\\(\\frac { 1 }{ 7 }\\),\\(\\frac { 1 }{ 8 }\\)}<\/strong>
\n (viii) {42, 49, 56, 63, 70, 77}<\/strong>
\nSolution:<\/strong><\/span>
\n(i)<\/strong> {3, 6, 9, 12, 15}
\n= {x: x is a natural number divisible by 3 ;x< 18}
\n(ii)<\/strong> {2, 3, 5, 7, 11, 13, }
\n= {x : x is a prime number}
\n(iii)<\/strong> {1,4,9, 16,25,36}
\n= {x : x is a perfect square ; x < 36}
\n(iv)<\/strong> {0, 2, 4, 6, 8, 10, 12, }
\n= {x : x is a whole number divisible by 2}
\n(v)<\/strong> {Monday, Tuesday, Wednesday}
\n= {x : x is one of the first three days of 3 week}
\n(vi)<\/strong> {23, 25, 27, 29, }
\n= {x : x is an odd natural number; x \u2265 23}
\n(vii)<\/strong> {\\(\\frac { 1 }{ 3 }\\),\\(\\frac { 1 }{ 4 }\\),\\(\\frac { 1 }{ 5 }\\),\\(\\frac { 1 }{ 6 }\\),\\(\\frac { 1 }{ 7 }\\),\\(\\frac { 1 }{ 8 }\\)}
\n= {x: x = \\(\\frac { 1 }{ n }\\) when n is a natural number: 3 \u2264 n \u2264 8}
\n(viii)<\/strong> {42, 49, 56, 63, 70, 77}
\n= (x: x is a natural number divisible by 7 ; 42 \u2264x \u2264 77}<\/p>\n
\nGiven : A = {x : x is a multiple of 2 and is less than 25}<\/strong>
\n B = {x : x is a square of a natural number and is less than 25}<\/strong>
\n C = {x : x is a multiple of 3 and is less than 25}<\/strong>
\n D = {x: x is a prime number less than 25}<\/strong>
\n Write the sets A, B, C and D in roster form.<\/strong>
\nSolution:<\/strong><\/span>
\nA = {x: x is a multiple of 2 and is less than 25}
\n= {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24}
\nB = {x : x is a square of natural number and is less than 25}
\n= {1,4,9,16}
\nC = {x : x is a multiple of 3 and is less than 25}
\n= {3, 6, 9, 12, 15, 18,21,24}
\nD = {x : x is a prime number less than 25}
\n= {2, 3, 5, 7, 11, 13, 17, 19, 23}<\/p>\nSet Concepts Exercise 13B –\u00a0Selina Concise Mathematics Class 7 ICSE Solutions<\/h3>\n
\nWrite the cardinal number of each of the following sets:<\/strong>
\n (i) A = Set of days in a leap year.<\/strong>
\n (ii) B = Set of numbers on a clopk-face.<\/strong>
\n (iii) C = {x : x \u2208 N and x \u2264 7}<\/strong>
\n (iv) D = Set of letters in the word \u201cPANIPAT\u201d.<\/strong>
\n (v) E = Set of prime numbers between 5 and 15.<\/strong>
\n (vi) F = {x : x \u2208 Z and – 2 < x \u2264 5}<\/strong>
\n (vii) G = {x : x is a perfect square number, x \u2208N and x \u2264 30}.<\/strong>
\nSolution:<\/strong><\/span>
\n(i)<\/strong> n A = 366
\n(ii)<\/strong> n B = 12
\n(iii)<\/strong> n C =7
\n(iv)<\/strong> n D = 5
\n(v)<\/strong> n E = 3
\n(vi)<\/strong> n F =7
\n(vii)<\/strong> n G = 5<\/p>\n
\nFor each set, given below, state whether it is finite set, infinite set or the null set :<\/strong>
\n (i) {natural numbers more than 100}<\/strong>
\n (ii) A = {x : x is an integer between 1 and 2}<\/strong>
\n (iii) B = {x : x \u2208 W ; x is less than 100}.<\/strong>
\n (iv) Set of mountains in the world.<\/strong>
\n (v) {multiples of 8}.<\/strong>
\n (vi) {even numbers not divisible by 2}.<\/strong>
\n (vii) {squares of natural numbers}.<\/strong>
\n (viii) {coins used in India}<\/strong>
\n (ix) C = {x | x is a prime number between 7 and 10}.<\/strong>
\n (x) Planets of the Solar system.<\/strong>
\nSolution:<\/strong><\/span>
\n(i)<\/strong> {Natural numbers more than 100}
\n= It is an infinite set
\n(ii)<\/strong> A = {x : x is an integer between 1 and 2}
\nIt is a null set
\n(iii)<\/strong> B = {x : x \u2208 W, x is less than 100}
\nIt is finite set as it has 100 elements i.e. from 0 to 99.
\n(iv)<\/strong> Set of mountains in the world.
\n\u2234 It is an infinite set
\n(v)<\/strong> {Multiples of 8}
\nIt is an infinite set
\n(vi)<\/strong> {Even numbers not divisible by 2}
\nIt is a null set
\n(vii)<\/strong> {Squares of natural numbers}
\n\u2234 It is an infinite set
\n(viii)<\/strong> {Coins used in India}
\n\u2234It is a finite set as these are countable
\n(ix)<\/strong> {x | x is a prime number between 7 and 10}
\nAs there is not such prime number between 7 and 10.
\nHence it is null set
\n(x)<\/strong> Planets of two Solar system.
\nIt is finite set as there are countable.<\/p>\n
\nState, which of the following pairs of sets are disjoint :<\/strong>
\n (i) {0, 1, 2, 6, 8}\u00a0<\/strong>and {odd numbers less than 10.<\/strong>
\n (ii) {birds} and {tress}<\/strong>
\n (iii) {x : x is a fan of cricket}\u00a0<\/strong>and {x : x is a fan of football}.<\/strong>
\n (iv) A = {natural numbers less than 10}\u00a0<\/strong>and B = {x : x is a multiple of 5}.<\/strong>
\n (v) {people living in Calcutta}\u00a0<\/strong>and {people living in West Bengal}.<\/strong>
\nSolution:<\/strong><\/span>
\n(i)<\/strong> {0, 1, 2, 6, 8} and {odd numbers less than 10}
\n\u21d2 {0, 1,2, 6, 8} and {1,3, 5, 7, 9}
\n\u2234There sets are not disjoint sets as there is one element (1) is common.
\n(ii)<\/strong> {Birds} and {trees}
\nThese are disjoint sets as there is no common element in term
\n(iii)<\/strong> {x : x is a fan of cricket} and {x : x is a fan of football}
\nThese are not disjoint sets as there can be a person who is fan of both the games.
\n(iv)<\/strong> A = {Natural numbers less than 10} and B = {x : x is a multiple of 5}
\n\u21d2 A = {1, 2, 3, 4, 5, 6, 7, 8, 9} and B = {5, 10, 15 }
\nThese are hot disjoint sets as there is one element 5, which is common.
\n(v)<\/strong> {People living in Calcutta} and {People living in West Bengal}.
\nThese are not disjoint sets as people of Calcutta are the people of West Bengal as Calcutta is a city of West Bengal.
\nSo, only (ii) is a pair of disjoint sets.<\/p>\n
\nState whether the given pairs of sets are equal or equivalent.<\/strong>
\n (i) A = {first four natural numbers} and B = {first four whole numbers}.<\/strong>
\n (ii) A = Set of letters of the word \u201cFOLLOW\u201d and B = Set of letters of the word \u201cWOLF\u201d.<\/strong>
\n (iii) E = {even natural numbers less than 10} and O = {odd natural numbers less than 9}<\/strong>
\n (iv) A = {days of the week starting with letter S} and B = {days of the week starting with letter T}.<\/strong>
\n (v) M = {multiples of 2 and 3 between 10 and 20} and N = {multiples of 2 and 5 between 10 and 20}.<\/strong>
\n (vi) P = {prime numbers which divide 70 exactly} and Q = {prime numbers which divide 105 exactly}<\/strong>
\n (vii) A = {0\u00b2, 1\u00b2, 2\u00b2, 3\u00b2, 4\u00b2} and = {16, 9,4, 1, 0}.<\/strong>
\n (viii) E = {8,JO, 12, 14, 16} and F = {even natural numbers between 6 and 18}.<\/strong>
\n (ix) A = {letters of the word SUPERSTITION} and\u00a0<\/strong>B = {letters of the word JURISDICTION}.<\/strong>
\nSolution:<\/strong><\/span>
\n(i)<\/strong> A = {first four natural numbers}
\n= {1,2, 3, 4}
\nB = {first first whole number)
\n= {0, 1,2,3}
\nThese are equivalent sets as both have equal number of elements but not same.
\n(ii)<\/strong> A = Set of letters of the word \u2018FOLLOW\u2019
\n= {F, O, L, W}
\nB = Set of letters of the word \u2018WOLF\u2019
\n= {W, O, L, F}
\nThese are equal sets as these have same and equal elements.
\n(iii)<\/strong> E = {even natural numbers less than 10}
\n= {2, 4, 6, 8}
\nO = {odd natural numbers less than 9}
\n= {L3, 5, 7}
\nThese are equivalent sets as both have equal number of elements but not the same.
\n(iv)<\/strong> A = {Days of the week starting with letter S}
\n= {Sunday, Saturday}
\nB = {Days of the week starting with letter T}
\n= {Tuesday, Thursday}
\nThese are equivalent sets as both have equal number of elements.
\n(v)<\/strong> M = {Multiples of 2 and 3 between 10 and 20}
\n= {12, 14, 15, 16, 18}
\nN = {Multiples of 2 and 5 between 10 and 20}
\n= {12, 14, 15, 16, 18}
\nThese are equal sets as these have same and equal number of elements.
\n(vi)<\/strong> P = {Prime numbers which divide 70 exactly}
\n= {2, 5, 7}
\nQ = {Prime numbers which divide 105 exactly}
\n= {3, 5, 7}
\nThese are equivalent sets as these have equal number of elements.
\n(vii)<\/strong> A = {02, l2, 22, 32, 42} = {0, 1, 4, 9, 16} B = {16, 9, 4, 1,0}
\nThese are equal sets as these have same and equal number of elements.
\n(viii)<\/strong> E = {8, 10, 12, 14, 16}