A) \[x=39\]
B) \[x=63\]
C) \[39\le x\le 63\]
D) None of these
Correct Answer: C
Solution :
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 \[n\,(A)=63,n\,(B)=76\] Now, \[n\,(A\cup B)=n(A)+n(B)-n(A\cap B)\] \[=63+76-n(A\cap B)\] \ \[n\,(A\cup B)+n(A\cap B)=139\] Þ \[n\,(A\cap B)=139-n(A\cup B)\] But \[n\,(A\cup B)\le 100\] \\[-n\,(A\cup B)\ge -100\] \ \[139-n\,(A\cup B)\ge 139-100=39\] \ \[n(A\cap B)\ge 39\] i.e., \[39\le n(A\cap B)\] .....(i) Again, \[A\cap B\subseteq A,A\cap B\subseteq B\] \ \[n\,(A\cap B)\le n\,(A)=63\] and \[n\,(A\cap B)\le n\,(B)=76\] \\[n(A\cap B)\le 63\] ?..(ii) Then, \[39\le n\,(A\cap B)\le 63\]Þ \[39\le x\le 63\].
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