Q.A market research group conducted a survey of 1000 consumers and reported that 720 consumers like product A and 450 consumers like product B. What is the least number that must have liked both the products?

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 $n(U) - n(A \cup B \cup C)$. …