Question:
A market research group conducted a survey of $2000$ consumers and reported that $1720$ consumers like product $P_1$ and $1450$ consumers like product $P_2$. What is the least number that must have liked both the products?
A market research group conducted a survey of $2000$ consumers and reported that $1720$ consumers like product $P_1$ and $1450$ consumers like product $P_2$. What is the least number that must have liked both the products?
Updated On: Jul 5, 2022
- $1150$
- $2000$
- $1170$
- $2500$
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The Correct Option is C
Solution and Explanation
Let $U$ be the set of all consumers who were questioned, $A$ be the set of consumers who liked product $P_1$ and $B$ be the set of consumers who liked the product $P_2$.
It is given that $n(U) =2000$, $n(A) = 1720$, $n(B) = 1450$.
$n(A \cup B) = n(A) + n(B) - n(A \cap B)$
$n(A \cup B) = 1720 + 1450 - n(A \cap B) = 3170 - n(A \cap B)$
Since, $A \cup B \subseteq U$
$\therefore n(A \cup B ) \le n(U)$
$\Rightarrow 3170 - n(A \cap B) \le 2000$
$\Rightarrow 3170-2000 \le n (A \cap B)$
$\Rightarrow n(A \cap B) \ge 1170$
Thus, the least value of $n(A \cap B)$ is $1170$.
Hence, the least number of consumers who liked both the products is $1170$.
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".