 Operations on Sets
Question

# 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

Easy
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 Now, $\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)$$=63+76-\mathrm{n}\left(\mathrm{A}\cap \mathrm{B}\right)$$\therefore \mathrm{n}\left(\mathrm{A}\cup \mathrm{B}\right)+\mathrm{n}\left(\mathrm{A}\cap \mathrm{B}\right)=139$$⇒\mathrm{n}\left(\mathrm{A}\cap \mathrm{B}\right)=139-\mathrm{n}\left(\mathrm{A}\cup \mathrm{B}\right)$But $\mathrm{n}\left(\mathrm{A}\cup \mathrm{B}\right)\le 100$$\therefore -\mathrm{n}\left(\mathrm{A}\cup \mathrm{B}\right)\ge -100$$\therefore 139-\mathrm{n}\left(\mathrm{A}\cup \mathrm{B}\right)\ge 139-100=39$Again, Then, $39\le \mathrm{n}\left(\mathrm{A}\cap \mathrm{B}\right)\le 63⇒39\le \mathrm{x}\le 63.$

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