Sets

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Sets

  1. 1. Class XI: MathsChapter 1: SetsKey Formulae1. Union of sets AB ={x:xA or xB }2. Intersection of sets AB ={x:xA and xB }3. Complement of a set A’ = {x: xU and xA},A’ = U-A4. Difference of sets A-B = {x: xA, xB} and B–A = {x: xB, xA}5. Properties of the Operation of Union.a. Commutative Law:A B = B Ab. Associative Law:(AB) C = A (BC)c. Law of IdentityA = Ad. Idempotent lawA A = Ae. Law of UU A = U6. Properties of Operation of Intersectioni) Commutative Law:A B = B Aii) Associative Law:(AB) C = A (BC)iii) Law of and UA =, U A = Uiv) Idempotent lawA A = Av) Distributive lawA (B C) = (A B) (A C)
  2. 2. 7. Properties of complement of sets: a. Complement laws: i. A A’ = U ii. A A’ =  b. De-Morgan’s law: i. (A B)’ = A’ B’ ii. (A B)’ = A’ B’ c. Law of double complementation: (A’)’ = A d. Laws of empty set and universal set: ’ = U and U’ =  8. Counting Theorems a. If A and B are finite sets, and A B = then number of elements in the union of two sets n(AUB) = n(A) + n(B) b. If A and B are finite sets, A B = then n(AU B ) = n(A) + n(B) - n(A ∩B) c. n(A B) = n(A – B) + n(B – A) + n(A B) d. n(A B  C) = n(A) + n(B) + n(C) – n(B∩C) – n(A∩B) – n(A∩C) + n(A∩B∩C) 9. Number of elements in the power set of a set with n elements =2n.Number of Proper subsets in the power set = 2n-2Question: Are the following pair of sets equal? Givereasons. (i) A = {2, 3}; B = {x: x is solution of x2 + 5x + 6 = 0}
  3. 3. (ii) A = {x: x is a letter in the word FOLLOW}; B = {y: y is a letter in the word WOLF} Answer (i) no [B={-2,-3}] (ii) yes. Question: Let A= {1, 2, {3, 4,}, 5}. Which of the following statements are incorrect and why? (i) {3, 4}⊂ A (ii) {3, 4}}∈ A (iii) {{3, 4}}⊂ A (iv) 1∈ A (v) 1⊂ A (vi) {1, 2, 5} ⊂ A (vii) {1, 2, 5} ∈ A (viii) {1, 2, 3} ⊂ A (ix) Φ ∈ A (x) Φ ⊂ A (xi) ,Φ- ⊂ A Answer F,T,T,T,F,F,F,F,F,T,FWhat is a Set?A set is a collection of discrete data items. The members ofthe set can be numbers or
  4. 4. names.Describing a SetThere are two distinct ways of describing the members of aset. One is to list themexplicitly, like you would find in a database of names.A = { Mark, Angela, Frank, Laura }A couple features of sets is that order doesn’t matter, andduplicates don’t really count.{ Mark, Angela, Frank, Laura } = { Laura, Frank, Mark, Angela}and { Mark, Angela, Frank, Laura } = { Mark, Laura, Angela, Mark, Frank, Laura }Another way to define a set is to describe a mathematicalrelationship.A = {x | 2x + 6 = 0 }The vertical bar can be read as “such that”, so that theentire statement would be read as“set A consists of members solving for x, such that 2 times xplus 6 equals 0”This same set can be listed explicitly. A = { -3 }x ∈ A (member / element of)Similarly, we use a slightly different symbol to state that thecontent of a variable is not amember of a particular set.x ∉ A (not a member / element of)This notation is good for individual members, but what ifwe are trying to compare a
  5. 5. group of set members? For that we have “subsets”. Asubset is any set whose membersare members of another set.A = { Mark, Angela }B = { Mark, Angela, Frank, Laura }Set A is a subset of set B because all members of set A are inset B. A symbol that iscommonly used is ⊆. Thus, we could writeA ⊆ B (subset)We make one additional distinction between sets, and thathas to do with whether everymember is accounted for. If every member is accounted for,the sets are equal. If they arenot, we have a proper subset. A proper subset is denotedusing a slightly differentsymbol.A ⊂ B (proper subset) Thus, if two sets are the same, then one cannot be a proper subset of the other.A power set is a collection (set) of sets which representsevery valid subset of a set. Thesymbol for the power set is a stylized P, or P. Thus, wherewe have a set…B = { Fred, Mary, Jane }The members of the power set for set B would be∅, {Fred}, {Mary}, {Jane}, {Fred, Mary},{Fred, Jane}, {Mary, Jane}, {Fred, Mary, Jane}We could also writeP (B) = { ∅, {Fred}, {Mary}, {Jane}, {Fred, Mary},
  6. 6. {Fred, Jane}, {Mary, Jane}, {Fred, Mary, Jane} }Notice that in the case above the number of elements in setB was 3. The number of elements in the power set of B is 8. The operation A – B removes thosemembers in set B that are in set A. If a member in set B isn’tin set A, then nothing isdone. (There is no sense of “negative data”, so you cannotremove what isn’t there.) Inour case we would get this:A = { Mary, Mark, Fred, Angela, Frank, Laura }B = { Fred, Mary, Frank, Jane } A – B = { Mark, Angela, Laura }Example Consider the setsφ, A = { 1, 3 }, B = {1, 5, 9}, C = {1, 3, 5, 7, 9}.Insert the symbol⊂ or⊄ between each of the following pairof sets:(i)φ . . . B(ii) A . . . B(iii) A . . . C(iv) B . . . CSolution (i)φ ⊂ B as φ is a subset of every set.(ii)A⊄ B as 3∈ A and 3∉ B(iii) A⊂ C as 1, 3∈ A also belongs to C(iv) B⊂ C as each element of B is also an element of C.
  7. 7. Example Let A = { a, e, i, o, u} and B = { a, b, c, d}. Is A asubset of B ? No.(Why?). Is B a subset of A? No. (Why?)Example L e t A, B and C be three sets. If A∈ B and B⊂ C, isit true thatA⊂ C? If not, give an example.Solution No. Let A = {1}, B = {{1}, 2} and C = {{1}, 2, 3}. HereA∈ B as A = {1}and B⊂ C. But A⊄ C as 1∈ A and 1∉ C.Note that an element of a set can never be a subset of itself.Subsets of set of real numbersThere are many important subsets of R. We give below the names of some of these subsets.The set of natural numbers N = {1, 2, 3, 4, 5, . . .}The set of integersZ = {. . ., –3, –2, –1, 0, 1, 2, 3, . . .}The set of rational numbers Q = {x :x = p/ q, p, q∈ Z and q ≠0}Question Let A ={1,2}, B={1,2,3,4}, c ={5,6}, D={5,6,7,8}, Verify that (a) AX(B∩C) = (AXB)∩(AXC) (b) AXC is a subsetof BXD.Answer take ordered pair and check the above results. Question: Show that the following four conditions areequivalent:(i) A ⊂ B (ii) A – B = Φ(iii) A ∪ B = B (iv) A ∩ B = A
  8. 8. Answer: (i)⬄(ii)⬄(iii)⬄(iv) as all elements of A are in B.Question: Is it true that for any sets A and B, P (A) ∪ P (B) =P (A ∪ B)? Justify your answer. Answer by an example A= {a},B= {b},above result is nottrue.Question: For any two sets A and B prove thatP(A)UP(B)⊂P(AUB) but , P(AUB) is not necessarily a subsetof P(A)UP(B). Answer Let X∈ P(A)UP(B)⇨ X⊂A or X⊂B ⇨ X ⊂AUB, forother part let A={1,2} and B={3,4,5} Then , we find X={1,2,3,4}⊂AUB, but X∉P(A),X∉P(B).SOX∉P(A)UP(B).Question: Using properties of sets show that(i) A ∪ (A ∩ B) = A (ii) A ∩ (A ∪ B) = AAnswer (i) (AUA)∩(AUB)=A∩(AUB)=A (ii) (A∩A)U(A∩B)=AU(A∩B)=AQuestion: Show that for any sets A and B,A = (A ∩ B) ∪ (A – B) and A ∪ (B – A) = (A ∪ B)Answer (A ∩ B) ∪ (A – B)= (A ∩ B) ∪ (A ∩ B’)=A∩X=A, A ∪ (B– A)=AU(B∩A’)=(AUB)∩(AUA’)
  9. 9. = (AUB)∩X=AUB.Question: If P(A)=P(B), show that A=B.Sol: Let x∈A ⇨ X∈P(A) ⇨X∈P(B) ⇨X⊂B⇨x∈B⇨A⊂B,similarily B⊂ A⇨A=B.Question: Let A,B and C be the three sets such thatAUB=AUC and A∩B=A∩C.Show that B=C.Answer ( AUB)∩C=(AUC )∩C (A∩C)U(B∩C) = (A∩C)U(C∩C) =CAgain, (AUB)∩B=(AUC)∩B (A∩B)U(B∩B) = (A∩B)U(C∩B) B = (A∩B)U(C∩B) ⇨B=C.A B
  10. 10. Union The union of two or more sets can be shown in adiagram as shown here. A’ A U A B =Notice that the contents of both circles are shaded. A∩B= A – B= A∩B’ C CAUBUC=
  11. 11. Example : A market research group conducted a survey of1000 consumers and reported that 720 consumers likeproduct A and 450 consumers like product B. What is theleast number that must have liked both products? Solution n(A ∩B)= 1170 – n(AUB), n(AUB)≤ n(U) then n(A∩B)≥170(least value) and maximum value of n(A ∩B) is1000.Example: A college awarded 38 medals in football, 15 inbasket ball and 20 in cricket. If These medals went to a total of 58 men and three men gotmedals in all three sports, how Many received medals in exactly two of three sports?Solution: n(F  = n(F) + n(B) + n(C) – n(B∩C) – n(F∩B) – B C)n(F∩C) +n(F∩B∩C)] or by venn diagram. 58 = 38+ 15 +20 - [ n(B∩C) + n(F∩B) + n(F∩C)]+3 18 = [ n(B∩C) + n(F∩B) + n(F∩C) ] Number of men who received medals in exactly two of thethree sports = n( F∩B∩C’)+( F∩B’∩C) + (F’∩B∩C) = n(B∩C) + n(F∩B) +n(F∩C) - 3 n(F∩B∩C)=9. ∵ n( F∩B∩C’) = n(F∩B) - n(F∩B∩C).
  12. 12. Question: In a survey of 60 people, it was found that 25people read newspaper H, 26 read newspaper T, 26 readnewspaper I, 9 read both H and I,11 read both H and T, 8read both T and I, 3 read all three newspapers. Find:(i) the number of people who read at least one of thenewspapers.(ii) the number of people who read exactly one newspaper. Answer (i) 52,(ii) 30 [n(A  = n(A) + n(B) + n(C) – B C) n(B∩C) – n(A∩B) – n(A∩C) + n(A∩B∩C)] or by venn diagram. Question: In a survey it was found that 21 people likedproduct A, 26 liked product B and 29 liked product C. If 14people liked products A and B, 12 people liked products Cand A, 14 people liked products B and C and 8 liked all thethree products. Find how many liked product C only,Product A and C but not product B , atleast one of threeproducts. Answer n(A’∩B’∩C’)=11, n(A∩B’∩C)=4 , n(AUBUC)=54. . **Question: Prove that for non-empty sets (AUBUC)∩(A∩B’∩C’)’∩C’ = B∩C’. Answer: L.H.S.⇨ (AUBUC)∩(A’UBUC)’∩C’ = (A∩A’)U(BUC)∩C’= (B∩C’)U(C∩C’)=R.H.S ∅ . **Question: Let A = {(x,y):y=ex ,x∈R} and B = {(x,y):y=e-x ,x∈R}. Is A∩B empty?
  13. 13. If not find the ordered pair belonging to A∩B. Answer: ex = e-x ⇨ e2x =1⇨ x=0, for x=0,y=1⇨ A and Bmeet on (0,1) and A∩B=∅. **Question: A and B are sets such that n(A-B)= 14+x, n(B-A)= 3x and n(A∩B)=x,draw venn diagram to illustrate theinformation and if n(A)=n(B), find x and n(AUB). Answer: n(A)=n(B) ⇨ n(A-B)+n(A∩B)=n(B-A)+n(A∩B)⇨x=7 n(AUB)= n(A-B)+n(A∩B)+n(B-A)=49.**Question: If A ={1}, find number of elements inP[P{P(A)}]. Answer: 16.**Question: Suppose A1,A2,A3....,A30 are thirty sets eachhaving 5 elements and B1,B2,B3,....Bn Are n sets each with 3 elements, let = =S and each element of S belongs to exactly 10 ofthe Ai’s and exactly 9 of the Bj’s.Then n is equal to ..... Answer: no. Of distinct elements in S= ==15= = =45.**Question: Two finite sets have m and n elements. Thenumber of subsets of the first Set is 112 more than that of the second set. Thevalues of m and n are, resp.(find) Answer: 2m-2n =112⇨ 2n(2(m-n) – 1)= 24(23 – 1).**Question: If X={8n – 7n – 1,n∈N} and Y = {49n – 49,n∈N}.Then find the relation b/w X,Y[X⊂ Y, Y⊂ X, X= Y, X ∩Y=∅] Answer: X=Y.[X= (1+7)n - 7n – 1,by binomialexpansion⇨ 49(C(n,2)+C(n,3)7+.....+C(n,n)7(n-2)) = 49(n-1)]
  14. 14. ASSIGNMENT(SETS) Question: 1 If U = {1,2,3......,10} , A = {1,2,3,5}, B = {2,4,6,7},then find (A-B)’. [Answer is {2,4,6,7...10} Question: 2 In an examination, 80% students passed inMathematics,72% passed in science and 13% failed in boththe subjects, if 312 students passed in both thesubjects.Find the total number of students who appeared inthe examination. [Answer number of students failed in both the subjects =n(M’∩S’)=13% of x=0.13x n(U) – n,(MUS)’- = 1.52x – 312 ⇨x=480.] Question: 3 If U = ,x :x ≤ 10, x∈ N}, A = {x :x ∈ N, x isprime}, B = {x : x∈ N, x is even} Write A ∩B’ in roster form. [ Answer is {3,5,7}] Question: 4 In a survey of 5000 people in a town, 2250were listed as reading English Newspaper, 1750 as readingHindi Newspaper and 875 were listed as reading both Hindias well as English. Find how many People do not read Hindior English Newspaper. Find how many people read onlyEnglish Newspaper.
  15. 15. Answer: People do not read Hindi or English Newspapern*(EUH)’+ = n(U) – n(EUH) = 1875, people read only EnglishNewspaper n(E’∩H) = n(E) – n(E∩ H) = 1375.Question 5 The Cartesian product AXA has 9 elementsamong which are found (-1,0) and (0,1). Find the set A and the remaining elements of AXA.Answer (-1,0) and (0,1)∈AXA ⇨ A = {-1,0,1} and AXA = {(-1,-1) ,( -1,0) , (-1,1) , (0,-1),(0,0) ,( 0,1) ,( 1,-1) , (1,0),(1,1)}Question 6 A and B are two sets such that n(A-B) = 14 + x, n(B-A) = 3x and n(A ∩B) =x. Draw the venn diagram to illustrate information and ifn(A) = n(B) then find the value of x.Answer n(A-B) = 14 + x= n(A ∩B’) = n(A) - n(A ∩B)⇨n(A)14+2x , n(B) = 4x ⇨ x=7 U A aaaaaaaaaaaaaaa B A-B B-A n(A∩B)
  16. 16. Question: 7 Let A and B be two sets , prove that: (A – B)UB = A iff B⊂A[Hint: (A ∩B’) U B=A ⇨ (AUB) ∩U =A⇨ B⊂ A If B⊂ A ,(A ∩B’) U B = (AUB) ∩U=A.] Question:8 In a survey of 100 students , the number ofstudents studying the various languages were found to be:English only 18,English but not Hindi 23,English and Sanskrit8, English 26, Sanskrit 48, Sanskrit and Hindi 8, no language24.Find: (i) How many students were studying Hind? (ii) How many students were studying English andHindi?[Hint: answer (i) 18 (ii) 3 , use venn diagram] Question: 9 In a survey of 500 television viewers producedthe following informations; 285 watch football, 195 watchhockey, 115 watch basketball, 45 watch football andbasketball, 70 watch football and hockey, 50 watch hockeyand basketball, 50 do not watch any of three games. Howmany watch all the three games? How many watch exactlyone of the three games? [Hint: answer 20 ,325]Question: 10 (i) Write roster form of {x: and 1≤ n ≤3 ,n∈ N}
  17. 17. (ii) Write set-builder form of {-4,-3,-2,-1,0,1,2,3,4}[ answer { ½,2/5,3/10} , {x: x∈Z , x2 <20} Question:11 If set A = {x:x=1/y, where y∈N},then which of the following belongs to A: 0, 1, 2, 2/3. [1] Question:12 If n(A) = 3, n(B) = 6 and the number of elements in AUB and in A∩B. Answer: A ⊆ B ⇨n(AUB)=n(B), n(A∩B)=n(A).

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