Question

#Quality Question
@Sets

A survey shows that 63% of Americans like cheese whereas 76% like Apples.
If x percent of Americans like both is and apples then . . .

1) x = 39
2)x = 63
3)39 ⩽ x ⩽63
4)None of these

Explanation Needed.​

Answer 1

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☆ Answer :

Option 3) 39 ⩽ x ⩽63

___________________________

☆ Given :

A Survey Concluded :-)

%of Americans who like CHEESE = 63%

% of Americans who like APPLES = 76%

% of Americans who like BOTH = x %

___________________________

☆ To Find :

% of American who like both i.e. value of x

___________________________

☆ Solution :

Let A and B denote the set of Americans who like Cheese and Apple respectively.

n(A) = 63 and n(B) = 76

According to De-Morgan's Law

n(A U B ) = n(A) + n(B) - n(A∩B)

⟹ n(A U B ) = 63 + 76 - n(A ∩ B)

⟹ n(A ∩ B) = 139 - n(A U B)

But, n(A U B) ⩽ 100

⟹ -n(A U B) ⩾ -100

⟹ 139 - n(A U B) ⩾ 139 - 100 = 39

⟹ 39 ⩽ n(A ∩ B) . . . (i)

Again ,

A ∩ B ⊆ A and A ∩ B ⊆ B

∴ n(A ∩ B) ⩽ n(A) = 63

and n(A ∩ B) ⩽ n(B) = 76

∴ n(A ∩ B) ⩽ 63 . . . (ii)

Combining eq (i) And (ii)

39 ⩽ n(A ∩ B) ⩽ 63

$\red{ \huge\underline{\boxed{ \bf 39 \leqslant x \leqslant 63} }}$

$\large\mathfrak{ \text{W}hich \: \: is \: \: the \: \: required \: \: \text{ A}nswer.}$

Answer 2

Soluťíòñ

Let Sets be such as

P = Americans who like Cheese

and

Q = Americans who like Apple

Accordingly,

n(P) = 63

and

n(Q) = 76

Now We know that ,

n(P U Q ) = n(P) + n(Q) - n(P∩Q) (De Morgan's Law)

n(P U Q ) = 63 + 76 - n(P ∩ Q)

n(P ∩ Q) = 139 - n(P U Q)

Also n(P U Q) ⩽ 100 (subset can exceed total no.present)

-n(P U Q) ⩾ -100 (while multiplying with -ve >>Sign of inequality changes )

139 - n(P U Q) ⩾ 139 - 100 = 39

∴ 39 ⩽ n(P ∩ Q)

Now ,

We Know

P ∩ Q ⊆ P and P ∩ Q ⊆ Q

(General Laws of Set Theory)

So,

n(P ∩ Q) ⩽ n(P) = 63

and n(P ∩ Q) ⩽ n(Q) ⩽ 76

∴ n(P ∩ Q) ⩽ 63

Both conditions Simultaneously concludes

39 ⩽ n(P ∩ Q) ⩽ 63

So,

39 ⩽ x ⩽63

Hence option 3 is correct.