Question

A market research group conducted a survey of $2000$ consumers and reported that $1720$ consumers liked product $P1$​ and $1450$ consumers liked product $P2$​. What is the least number that must have liked both the products.

Answer 1

We are given that, a market research group conducted a survey of $2000$ consumers and reported that $1720$ consumers liked product $P1$​ and $1450$ consumers liked product $P2$​. We can say that, $N(P1\cup P2)$ $=2000$, $N(P1)$ $=1720$ and $N(P2)$ $=1450$ and we have to find the least number that must have liked both the products i.e. $N(P1\cap P2)$. Here, use the identity $N(A\cup B)=N(A)+N(B)-N(A\cap B)$ and solve it.

Complete step-by-step answer:
We are given that, a market research group conducted a survey of $2000$ consumers and reported that $1720$ consumers liked product $P1$​ and $1450$ consumers liked product $P2$​.
Here, total number of consumers $=N(P1\cup P2)$ $=2000$
Number of consumers liking $N(P1)$ $=1720$
Also, number of consumers liking $N(P2)$ $=1450$
Now we have to find the least number that must have liked both the products i.e. $N(P1\cap P2)$.
Now we know the identity that,
$N(A\cup B)=N(A)+N(B)-N(A\cap B)$
Here we have, $A=P1$ and $B=P2$.
Now we get,
$N(P1\cup P2)=N(P1)+N(P2)-N(P1\cap P2)$
Now substituting the given values in above we get,
$2000=1720+14500-N(P1\cap P2)$
Simplifying in simple form we get,
$\Rightarrow$ $N(P1\cap P2)=1720+1450-2000$
$\Rightarrow$ $N(P1\cap P2)=3170-2000$
Again, simplifying we get,
$\Rightarrow$ $N(P1\cap P2)=1170$
Therefore, we get the least number that must have liked both the products is $1170$.

Additional information:
The algebraic equations which are valid for all values of variables in them are called algebraic identities. They are also used for the factorization of polynomials. Sets, in mathematics, are an organized collection of objects and can be represented in set-builder form or roster form. Sets are represented as a collection of well-defined objects or elements and it does not change from person to person. A set is represented by a capital letter. The number of elements in the finite set is known as the cardinal number of a set.

Note: Here we are given the total number of consumers and number of consumers in each term. Most of the time the mistake or confusion occurs between union and intersection. There is confusion between these two.
If set $A$ and set $B$ are two sets, then $A$ union $B$ is the set that contains all the elements of set $A$ and set $B$. It is denoted as $A\cup B$.
If set $A$ and set $B$ are two sets, then $A$ intersection $B$ is the set that contains only the common elements between set $A$ and set $B$. It is denoted as $A\cap B$.