Question

There are 200 individuals with a skin disorder, 120 have been exposed to chemical ${{C}_{1}}$, 50 to chemical ${{C}_{2}}$ and 30 to both the chemicals. Find the number of individuals exposed to (i) chemical ${{C}_{1}}$ or chemical ${{C}_{2}}$ (ii) Chemical ${{C}_{1}}$ but not chemical ${{C}_{2}}$ (iii) chemical ${{C}_{2}}$ but not chemical ${{C}_{1}}$.

Answer 1

Hint: You need to solve this question by understanding the meaning of different operations between two sets. You also need to use the data that there are a total of two sets present while drawing the Venn diagram.

Complete step-by-step solution -

Before we start with the solution, let us understand the meaning of different symbols and terms used in the question.

Universal set: The set containing all objects or elements and of which all other sets are subsets. So, for our question, the universal set is the union of the three sets A, B, and C.

Union: The union (denoted by $\cup$ ) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other.

Intersection: The intersection of two sets has only the elements common to both sets. If an element is in just one set, it is not part of the intersection. The symbol is an upside down $\cap$ .

Now, let us start with the actual solution to the question given above by drawing the Venn diagram of the situation given above.

Now according to the question, ${{C}_{1}}\cup {{C}_{2}}=200,\text{ }{{\text{C}}_{1}}=120\text{, }{{\text{C}}_{2}}=50,\text{ }{{C}_{1}}\cap {{C}_{2}}=30$ .

Now we will solve the different subparts of the question.

(i). Chemical ${{C}_{1}}$ or chemical ${{C}_{2}}$ = $\left( {{C}_{1}}\cup {{C}_{2}} \right)-\left( {{C}_{1}}\cap {{C}_{2}} \right)=170$

(ii). Chemical ${{C}_{1}}$ but not chemical ${{C}_{2}}$ = ${{C}_{1}}-\left( {{C}_{1}}\cap {{C}_{2}} \right)=90$

(iii). Chemical ${{C}_{2}}$ but not chemical ${{C}_{1}}$ = ${{C}_{2}}-\left( {{C}_{1}}\cap {{C}_{2}} \right)=20$

Note: Be careful about the symbols of union and intersection, as they might be confusing. Also, be careful and cross-check the set notations of each case, as it’s the only place where mistakes can be committed.