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# 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).