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A survey shows that 63% of the Americans like cheese whereas 76% like apples. If the x% of the Americas like both cheese and apples , find the value of x.
n(C U A) = n(A) + n(C) - n(C n A)100 = 76 + 63 - X100 = 139 - XX = 139 - 100X = 39sher mohammadB.Tech, IIT Delhi
n(C U A) = n(A) + n(C) - n(C n A)
100 = 76 + 63 - X
100 = 139 - X
X = 139 - 100
X = 39
sher mohammad
B.Tech, IIT Delhi
You can tell like that total % of union of set C and A can only be 100 that is why we take n(c U a) = 100
Let A denotes, the set of Americans who like cheese andB denotes those who like apples. Let the population of America be 100, thenn(A)=63andn(B)=76n(A)=63andn(B)=76Now, n(A∪B)=n(A)+n(B)−n(A∩B)n(A∪B)=n(A)+n(B)−n(A∩B)N(A∪B)=63+76−n(A∩B)N(A∪B)=63+76−n(A∩B)N(A∩B)=139−n(A∪B)N(A∩B)=139−n(A∪B)But n(A∪B)≤100n(A∪B)≤100So, 139−n(A∪B)≥139−100139−n(A∪B)≥139−100139−n(A∪B)≥39139−n(A∪B)≥39n(A∩B)≥39n(A∩B)≥39…………(i)Now,n(A∩B)≤n(A)andn(A∩B)≤n(B)n(A∩B)≤n(A)andn(A∩B)≤n(B)n(A∩B)≤63andn(A∩B)≤76n(A∩B)≤63andn(A∩B)≤76n(A∩B)≤63n(A∩B)≤63……………(ii)From Eqs. (i) and (ii),39≤n(A∩B)≤6339≤n(A∩B)≤6339≤x≤63
Let A denotes, the set of Americans who like cheese andB denotes those who like apples. Let the population of America be 100, then n(A)=63andn(B)=76n(A)=63andn(B)=76Now, n(A∪B)=n(A)+n(B)−n(A∩B)n(A∪B)=n(A)+n(B)−n(A∩B)N(A∪B)=63+76−n(A∩B)N(A∪B)=63+76−n(A∩B)N(A∩B)=139−n(A∪B)N(A∩B)=139−n(A∪B)But n(A∪B)≤100n(A∪B)≤100So, 139−n(A∪B)≥139−100139−n(A∪B)≥139−100139−n(A∪B)≥39139−n(A∪B)≥39n(A∩B)≥39n(A∩B)≥39…………(i)Now,n(A∩B)≤n(A)and n(A∩B)≤n(B)n(A∩B)≤n(A)and n(A∩B)≤n(B)n(A∩B)≤63and n(A∩B)≤76n(A∩B)≤63and n(A∩B)≤76n(A∩B)≤63n(A∩B)≤63……………(ii)From Eqs. (i) and (ii),39≤n(A∩B)≤6339≤n(A∩B)≤6339≤x≤63Cheers !!
n(C U A) = n(A) + n(C) - n(C n A)100 = 76 + 63 - X100 = 139 - XX = 139 - 100X = 39 You can tell like that total % of union of set C and A can only be 100 that is why we take n(c U a) = 100
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