A survey shows that $$63$$% of the Americans like cheese whereas $$76$$% like apples. If x% of the Americans like both cheese and apples, then

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Infinity Learn · Accepted Answer

Let A denote the set of Americans who like cheese and let B denote the set of Americans who like apples.Let Population of American be $$100$$.Then $$n(A)=63$$, $$n(B)=76Now$$, $$n(A$$∪$$B)=n(A)+n(B)-n(A$$∩$$B)=63+76-n(A$$∩$$B)$$∴$$n(A$$∪$$B)+n(A$$∩$$B)=139$$⇒$$n(A$$∩$$B)=139-n(A$$∪$$B)But$$ $$n(A$$∪$$B)$$≤$$100$$∴$$-n(A$$∪$$B)$$≥$$-100$$∴$$139-n(A$$∪$$B)$$≥$$139-100=39$$∴$$n(A$$∩$$B)$$≥$$39$$ i.e., $$39$$≤$$n(A$$∩$$B)$$    ...(i)Again$$, A∩B⊆A, A∩B⊆B∴$$n(A$$∩$$B)≤$$n(A)=63$$ and $$n(A$$∩$$B)$$≤$$n(B)=76$$∴$$n(A$$∩$$B)$$≤$$63$$    ...(ii)Then$$, $$39$$≤$$n(A$$∩$$B)≤$$63$$⇒$$39$$≤x≤$$63$$.

A survey shows that 63% of the Americans like cheese whereas 76% like apples. If x% of the Americans like both cheese and apples, then

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