A) \[39\le x\le 63\]
B) \[63\]
C) \[39\]
D) \[139\ge x\]
Correct Answer: A
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, \[n(A\cup B)=n(A)+n(B)-n(A\cap B)\] \[=63+76-n(A\cap B)\] \[\Rightarrow \] \[n(A\cup B)+n(A\cap B)=139\] \[\Rightarrow \] \[n(A\cap B)=139-n(A\cup B)\] But \[n(A\cup B)\le 100\] \[\therefore \] \[-n(A\cup B)\ge -100\] \[\Rightarrow \] \[139-n(A\cup B)\ge 139-100=39\] \[\Rightarrow \]\[n(A\cap B)\ge 39\]i.e.,\[39\le n(A\cap B)\] ?.(i) Again, \[A\cap B\subseteq A,\,A\cap B\subseteq B\] \[\therefore \] \[n(A\cap B)\,\,\le \,n(A)=63\] and \[n(A\cap B)\,\,\le \,n(B)=76\] \[\therefore \] \[n(A\cap B)\le 63\] ?..(ii) Then, \[39\,\,\le n(A\cap B)\le \,63\] [from Eq. (i) and (ii)] \[\Rightarrow \] \[39\le x\le 63\]
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