Venn Diagrams for CSAT: Two-Set and Three-Set Problems

A Venn diagram is a picture that shows how two or more groups overlap. Each group is drawn as a closed curve, usually a circle, and the part where the circles overlap stands for the members that belong to both groups at once. For CSAT Paper II, which is a qualifying paper (you need roughly 33 percent to clear it, and these marks do not add to your merit rank), Venn diagram questions are among the cheapest marks on offer: almost every one of them is a single idea, the inclusion-exclusion principle, dressed up in words about newspapers, magazines, students or employees.

What a Venn diagram actually shows

Imagine a rectangle that contains everyone surveyed: this is the universal set. Inside it, draw one circle for everyone who reads newspaper A and another circle for everyone who reads newspaper B. The two circles divide the rectangle into four regions: people who read only A, people who read only B, people who read both A and B (the lens-shaped overlap), and people outside both circles who read neither. Every Venn question is just a question about filling in these four regions.

Take a quick numerical example to fix the picture. Suppose 100 people are surveyed: 60 read newspaper A, 50 read newspaper B, and 30 read both. The overlap (both) is 30, so only A is 60 minus 30 = 30, only B is 50 minus 30 = 20, and neither is 100 minus (30 + 30 + 20) = 20. Notice that 60 + 50 = 110 is more than 100: the 30 people in the overlap were counted twice, once in each circle, and the whole subject is about correcting that double counting.

The two-set rule: add, then subtract the overlap

The inclusion-exclusion principle for two sets says: the number in at least one of the two groups equals the sum of the two groups minus the number in both, because the overlap was included twice. In symbols, n(A union B) = n(A) + n(B) minus n(A intersection B), where n(A union B) means everyone in A or B (or both), and n(A intersection B) means everyone in both. The people in neither group are simply the total minus this union.

Worked example (UPSC CSE 2018). 19 boys turned out to play hockey. Of these, 11 wore hockey shirts and 14 wore hockey pants, and every boy wore at least one of the two. How many wore the full uniform (both)? Step 1: at least one = 19 (every boy is covered). Step 2: shirts + pants = 11 + 14 = 25. Step 3: the overlap = 25 minus 19 = 6. So 6 boys wore both the shirt and the pants. Whenever the stem says every member belongs to at least one group, the union is the total itself.

Worked example (UPSC CSE 2024). 80 percent passed English, 70 percent passed Hindi, and 15 percent failed in both subjects. What percentage failed in only one subject? First flip the failed-in-both figure: 15 percent failed both means 85 percent passed at least one subject. Next, both-subjects passers = 80 + 70 minus 85 = 65 percent. The passers in exactly one subject = at least-one (85) minus both (65) = 20 percent. Since failing in exactly one subject is the same group as passing in exactly one, the answer is 20 percent.

Three-set problems: add, subtract, add back

With three groups the same logic stretches one step further. Add the three single counts; the three pairwise overlaps were each counted twice, so subtract each of them; but the people in all three groups were subtracted three times too many, so add them back once. The formula is: n(A union B union C) = (A + B + C) minus (pairwise overlaps) plus (all three). The trick that saves time is to always fill the diagram from the inside out: write the all-three number in the centre first, then the only-two regions, then the only-one regions.

Worked example (UPSC CSE 2015). In a town, 45 percent read magazine A, 55 percent read B, 40 percent read C; 30 percent read A and B, 15 percent read B and C, 25 percent read A and C; 10 percent read all three. What percentage reads no magazine? At least one magazine = (45 + 55 + 40) minus (30 + 15 + 25) plus 10 = 140 minus 70 plus 10 = 80 percent. So the percentage that reads none = 100 minus 80 = 20 percent. Note carefully: each pairwise figure (say 30 percent for A and B) already includes the 10 percent who read all three, which is exactly why the formula works.

Minimum and maximum overlap: when the overlap is not given

Sometimes the question gives the two group sizes but not their overlap, and asks how many must belong to both. Here the overlap can range between two extremes. The maximum overlap is the size of the smaller group (everyone in the smaller group could sit inside the larger one). The minimum overlap is n(A) + n(B) minus the total, floored at zero: if the two groups together are bigger than the whole population, the excess must be sitting in the overlap.

Worked example (UPSC CSE 2021). In a group of 120 persons, 80 are Indians and 70 can speak English. How many Indians can speak English? Maximum overlap = the smaller of 80 and 70 = 70 (all English speakers could be Indians). Minimum overlap = 80 + 70 minus 120 = 30 (the two groups together exceed the total by 30, so at least 30 Indians speak English). The count therefore lies between 30 and 70, which matches option (d): 30 or more. Whenever UPSC asks for the number in both without fixing it, expect a range and pick the option that describes the range.

Formula cheat sheet

Traps UPSC sets

• Anchor the total correctly. In the hockey question the 19 boys were the whole universe, not a fraction of a bigger class; do not invent a larger total than the stem gives.
• Only versus at least one. Only A means the circle-minus-lens region; at least one means the whole of both circles. UPSC deliberately phrases options around this mix-up.
• Pairwise figures include the triple overlap. 30 percent read A and B already contains the 10 percent who read all three; never subtract the centre again when using the formula.
• Flip failed and passed correctly. 15 percent failed both subjects means 85 percent passed at least one; failing exactly one subject is the same group as passing exactly one.
• Percentages are counts out of 100. When a stem uses percentages, quietly set the total to 100 and treat every percent as a headcount; the arithmetic becomes addition and subtraction.

Speed tactics

• Draw tiny circles, or just boxes. You do not need art; two overlapping ellipses with four labelled regions, filled from the inside out, beat pure algebra every time.
• Compute the union first. Almost every two-set question reduces to one line: union = A + B minus both; everything else (only, neither, exactly one) falls out of it.
• For three sets, start at the centre. Write the all-three number in the middle first; each only-two region is its pair figure minus the centre; each only-one region is its circle minus everything else in it.
• Use the extremes when the overlap is missing. If the stem fixes only the two group sizes, compute max = smaller group and min = A + B minus total, then match the option describing that range.

Key Terms

• Set: a collection of distinct objects or people sharing a property, such as all students who passed English.
• Element (member): a single object inside a set; in Venn questions, one person surveyed.
• Universal set: the rectangle in a Venn diagram: everyone or everything under consideration, from which nothing is excluded.
• Union (A union B): everyone in A, or in B, or in both; the whole of both circles.
• Intersection (A intersection B): everyone in both A and B at once; the lens-shaped overlap of the circles.
• Complement (not A): everyone in the universal set who is not in A; everything outside circle A.
• Only A: members of A excluding anyone also in B; circle A minus the overlap region.
• Neither: members outside all the circles; the universal set minus the union of the groups.
• Disjoint sets: two sets with no overlap at all; their intersection is empty, so the union is just the sum.
• Inclusion-exclusion principle: the counting rule that adds group sizes and then subtracts the overlaps to correct double counting.
• Cardinality: the number of elements in a set, written n(A); the quantity every Venn formula manipulates.

Practice questions