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If-n-A-15-and-n-B-25-a-What-are-the-greatest-and-least-values-of-n-AuB-b-What-are-the-greatest-and-least-value-of-n-AnB-c-Draw-Venn-diagrams-to-illustrate-the-four-situations-in-

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Question Number 28806 by pieroo last updated on 30/Jan/18
If n(A)=15 and n(B)=25, (a) What are the greatest and least values of n(AuB)? (b) What are the greatest and least value of n(AnB)? (c) Draw Venn diagrams to illustrate the four situations in (a) and (b) above
$$\mathrm{If}\:\mathrm{n}\left(\mathrm{A}\right)=\mathrm{15}\:\mathrm{and}\:\mathrm{n}\left(\mathrm{B}\right)=\mathrm{25},\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{What}\:\mathrm{are}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{and}\:\mathrm{least}\:\mathrm{values}\:\mathrm{of}\:\mathrm{n}\left(\mathrm{AuB}\right)? \\ $$$$\left(\mathrm{b}\right)\:\mathrm{What}\:\mathrm{are}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{and}\:\mathrm{least}\:\mathrm{value}\:\mathrm{of}\:\mathrm{n}\left(\mathrm{AnB}\right)? \\ $$$$\left(\mathrm{c}\right)\:\mathrm{Draw}\:\mathrm{Venn}\:\mathrm{diagrams}\:\mathrm{to}\:\mathrm{illustrate}\:\mathrm{the}\:\mathrm{four}\: \\ $$$$\:\:\:\:\:\:\:\mathrm{situations}\:\mathrm{in}\:\left(\mathrm{a}\right)\:\mathrm{and}\:\left(\mathrm{b}\right)\:\mathrm{above} \\ $$
Answered by Rasheed.Sindhi last updated on 31/Jan/18
Formula: n(A∪B)=n(A)+n(B)−n(A∩B) The above formula suggest that: ^• n(A∪B) is the greatest when n(A∩B) is the least i-e n(A∩B)=0 or A∩B=φ i-e A & B are disjoint sets. n(A∪B)=n(A)+n(B)−n(A∩B) =15+25−0=40 ^• n(A∪B) is the least when n(A∩B) is the greatest. n(A∩B) is the greatest when A⊂B .In this case n(A∩B)=15 ∴ n(A∪B)=15+25−15=25 (a)The greatest value of n(A∩B)=40 The least value of n(A∩B)=25 −.−.− (b)The gretest n(A∩B)=15 The least n(A∩B)=0
$$\mathrm{Formula}: \\ $$$$\:\:\mathrm{n}\left(\mathrm{A}\cup\mathrm{B}\right)=\mathrm{n}\left(\mathrm{A}\right)+\mathrm{n}\left(\mathrm{B}\right)−\mathrm{n}\left(\mathrm{A}\cap\mathrm{B}\right) \\ $$$$\mathrm{The}\:\mathrm{above}\:\mathrm{formula}\:\mathrm{suggest}\:\mathrm{that}: \\ $$$$\:^{\bullet} \mathrm{n}\left(\mathrm{A}\cup\mathrm{B}\right)\:\mathrm{is}\:\mathrm{the}\:\boldsymbol{\mathrm{greatest}}\:\mathrm{when}\:\mathrm{n}\left(\mathrm{A}\cap\mathrm{B}\right)\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{least}\:\mathrm{i}-\mathrm{e}\:\mathrm{n}\left(\mathrm{A}\cap\mathrm{B}\right)=\mathrm{0}\:\mathrm{or}\:\mathrm{A}\cap\mathrm{B}=\phi \\ $$$$\mathrm{i}-\mathrm{e}\:\mathrm{A}\:\&\:\mathrm{B}\:\mathrm{are}\:\mathrm{disjoint}\:\mathrm{sets}. \\ $$$$\:\:\:\:\mathrm{n}\left(\mathrm{A}\cup\mathrm{B}\right)=\mathrm{n}\left(\mathrm{A}\right)+\mathrm{n}\left(\mathrm{B}\right)−\mathrm{n}\left(\mathrm{A}\cap\mathrm{B}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{15}+\mathrm{25}−\mathrm{0}=\mathrm{40} \\ $$$$\:^{\bullet} \mathrm{n}\left(\mathrm{A}\cup\mathrm{B}\right)\:\mathrm{is}\:\mathrm{the}\:\boldsymbol{\mathrm{least}}\:\mathrm{when}\:\mathrm{n}\left(\mathrm{A}\cap\mathrm{B}\right)\:\mathrm{is} \\ $$$$\mathrm{the}\:\boldsymbol{\mathrm{greatest}}.\:\mathrm{n}\left(\mathrm{A}\cap\mathrm{B}\right)\:\mathrm{is}\:\mathrm{the}\:\mathrm{greatest} \\ $$$$\mathrm{when}\:\mathrm{A}\subset\mathrm{B}\:.\mathrm{In}\:\mathrm{this}\:\mathrm{case}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{n}\left(\mathrm{A}\cap\mathrm{B}\right)=\mathrm{15} \\ $$$$\therefore\:\:\mathrm{n}\left(\mathrm{A}\cup\mathrm{B}\right)=\mathrm{15}+\mathrm{25}−\mathrm{15}=\mathrm{25} \\ $$$$\left(\mathrm{a}\right)\mathrm{The}\:\mathrm{greatest}\:\mathrm{value}\:\mathrm{of}\:\mathrm{n}\left(\mathrm{A}\cap\mathrm{B}\right)=\mathrm{40} \\ $$$$\mathrm{The}\:\mathrm{least}\:\mathrm{value}\:\mathrm{of}\:\mathrm{n}\left(\mathrm{A}\cap\mathrm{B}\right)=\mathrm{25} \\ $$$$−.−.− \\ $$$$\left(\mathrm{b}\right)\mathrm{The}\:\mathrm{gretest}\:\mathrm{n}\left(\mathrm{A}\cap\mathrm{B}\right)=\mathrm{15} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{The}\:\mathrm{least}\:\mathrm{n}\left(\mathrm{A}\cap\mathrm{B}\right)=\mathrm{0} \\ $$

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