Q.In a group of 400 people, who speak either Hindi or English or both, if 230 speak Hindi only and 70 speak both English and Hindi, then how many of them speak English only?

Practical Problems on Operations on Sets

You already know what a set is — a collection of distinct objects. And you know the basic operations: union ($A \cup B$), intersection ($A \cap B$), difference ($A - B$), and complement ($A'$). The question is: why do we need to solve problems with them? Because in real life, data rarely comes neatly labelled as "these are the elements of set A". Instead, you get a paragraph about a survey, or a group of people, or a list of items with overlapping properties. Your job is to translate that messy reality into set language, apply the operations, and extract the answer.

The core idea: counting with overlaps

Imagine a class of 30 students. 18 play cricket, 15 play football, and 10 play both. How many play at least one sport? How many play neither?

If you simply add 18 + 15, you get 33 — but there are only 30 students. The mistake is that the 10 students who play both have been counted twice. The union formula fixes this:

Here, $n(A \cup B) = 18 + 15 - 10 = 23$. So 23 students play at least one sport. The remaining $30 - 23 = 7$ play neither.

The general formula for three sets

When three sets overlap, the same logic extends. For sets $A$, $B$, $C$:

Why the plus at the end? Because when you subtract all three pairwise intersections, the triple intersection (the region common to all three) gets subtracted three times — but it was originally added three times. So it's been removed entirely. Adding it back once corrects that.

How to approach any problem

1. Identify the universal set — the whole group under consideration. Its size is $n(U)$.
2. Define your sets clearly. Usually the problem gives you categories: "people who like tea", "people who like coffee", etc.
3. Extract the given numbers — total, individual set sizes, pairwise intersections, triple intersection.
4. Decide what is being asked — union? complement? exactly one? exactly two?
5. Apply the appropriate formula or draw a Venn diagram.

Venn diagrams: your visual tool

For problems with two or three sets, a Venn diagram is often faster than memorising formulas. Draw three overlapping circles inside a rectangle (the universal set). Label each region. Fill in known numbers from the innermost region outward — start with the triple intersection, then the pairwise overlaps (subtracting the triple), then the individual sets (subtracting the overlaps). Finally, the region outside all circles is $n(U) - n(A \cup B \cup C)$.

A worked example

In a survey of 100 people: 60 like tea, 50 like coffee, 30 like both. How many like neither?

• $n(U) = 100$, $n(T) = 60$, $n(C) = 50$, $n(T \cap C) = 30$
• $n(T \cup C) = 60 + 50 - 30 = 80$
• $n(\text{neither}) = 100 - 80 = 20$

So 20 people like neither tea nor coffee.

What about "exactly one" or "exactly two"?

These are common variations. For two sets, "exactly one" means $n(A) + n(B) - 2n(A \cap B)$ — because the intersection is counted in both $n(A)$ and $n(B)$, so subtracting it twice leaves only the parts that belong to exactly one set.

For three sets, "exactly two" means the sum of the three pairwise intersections minus three times the triple intersection (since the triple is counted in each pairwise intersection). "Exactly one" is the union minus the "exactly two" and "exactly three" regions.

The bottom line

Practical problems on sets are just about translating words into set notation, then using the inclusion-exclusion principle (the union formula) to count correctly. The formulas are simple, but the trap is always in the reading — misidentifying what "both", "either", "neither", "exactly one" mean in the given context. Draw the diagram, label carefully, and the numbers will fall into place.

The final answer is always a number — but the real skill is in the translation.