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Probability
Every CAT Probability formula on one page — conditional probability, Bayes' theorem, binomial probability, and expected value — each with a worked example.
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Set Theory
CAT'26 QUANT CHEATSHEET
Every set theory formula and CAT shortcut you need for CAT 2026 — on one page.
Set Theory on CAT rarely tests the abstract definitions directly — it shows up as a word problem about how many students study exactly two subjects, or how many people like coffee but not tea. Underneath the story, every question reduces to the same Venn-diagram cardinality formulas and a careful read of which region is actually being asked for. This sheet lays out every formula you need for the CAT quant section: basic notation, subsets and power sets, union, intersection and complement, De Morgan’s laws, two- and three-set cardinality, only-one-set and exactly-two-set regions, and the Cartesian product, each with a worked example in real numbers. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is costing you marks.
Set Theory: every formula you need
1Basic Set Notation
A set is a well-defined collection of distinct objects.
x ∈ A means x belongs to A | x ∉ A means x does not
Example: A={1,2,3,4,5} → 3∈A, but 6∉A.
2Types of Sets
The special sets that recur across every set-theory question.
empty ∅ (0 elements) | singleton (1) | universal U (everything under discussion)
Example: {x : x is a day of the week starting with ‘Z’} = ∅.
CAT Hack: The universal set U depends entirely on context — always pin down what U is before working the problem.
3Subsets & Power Set
Every possible subset of a set, counted in one formula.
A ⊆ B: every element of A is in B | number of subsets = 2n
Example: A={1,2,3} → 23 = 8 subsets (including ∅ and A itself).
CAT Favourite: The 2ⁿ subset count is a recurring CAT shortcut — remember it always includes the empty set and the full set.
4Union & Intersection
The two basic ways to combine two sets.
A ∪ B = elements in A or B (or both) | A ∩ B = elements in both
Example: A={1,2,3}, B={2,3,4} → A∪B={1,2,3,4}, A∩B={2,3}.
5Complement of a Set
Everything in the universal set that is not in A.
A′ = U − A
Example: U={1,…,10}, A={2,4,6,8,10} → A′={1,3,5,7,9}.
CAT Insight: A∩A′=∅ and A∪A′=U always hold — a set and its complement partition the universal set.
6Difference & Symmetric Difference
Elements in only one set, versus elements in exactly one of two.
A−B = in A but not B | AΔB = (A−B)∪(B−A)
Example: A={1,2,3}, B={2,3,4} → A−B={1}, AΔB={1,4}.
Common Mistake: A−B and B−A are not the same set — order matters, unlike union or intersection.
7Venn Diagram: Two Sets
The core cardinality formula for two overlapping sets.
n(A∪B) = n(A) + n(B) − n(A∩B)
Example: n(A)=30, n(B)=20, n(A∩B)=10 → n(A∪B) = 40.
8Venn Diagram: Three Sets
The same idea extended to three overlapping sets.
n(A∪B∪C) = n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(A∩C)+n(A∩B∩C)
Example: 40,35,30 with pairwise 15,10,8 and triple 5 → 40+35+30−15−10−8+5 = 77.
CAT Hack: Sketch the Venn diagram and fill the innermost (triple-overlap) region first, then work outward — it avoids sign errors.
9De Morgan's Laws
Complementing a union or intersection flips the operation too.
(A∪B)′ = A′∩B′ | (A∩B)′ = A′∪B′
Example: U={1..10}, A∪B={1,2,3,4} → (A∪B)′ = {5,…,10} = A′∩B′.
CAT Insight: Complementing doesn't just flip ∪ to ∩ (or vice versa) — every individual set inside also gets complemented.
10Cardinality Formula, Rearranged
Solve for any missing term in the two-set union formula.
n(A∩B) = n(A) + n(B) − n(A∪B)
Example: 60 people, 35 like tea, 30 like coffee, all like at least one → n(A∩B) = 35+30−60 = 5.
11Only-One-Set Regions
The part of a set that doesn't overlap with the other.
only A = n(A) − n(A∩B) | only B = n(B) − n(A∩B)
Example: n(A)=30, n(B)=20, n(A∩B)=10 → only A=20, only B=10.
Common Mistake: “Only A” is not the same as n(A) — you must subtract the overlap first.
12Set Identities
The distributive laws connecting union and intersection.
A∩(B∪C) = (A∩B)∪(A∩C) | A∪(B∩C) = (A∪B)∩(A∪C)
Example: A={1,2}, B={2,3}, C={1,3} → A∩(B∪C) = {1,2} = (A∩B)∪(A∩C).
13Disjoint Sets & Partitions
Sets with nothing in common, and a way to split a whole set exactly.
disjoint: A∩B=∅ | partition: pairwise disjoint sets whose union is the whole set
Example: {1,2,3} and {4,5,6} are disjoint, and together partition {1,…,6}.
14Cartesian Product
Every possible ordered pair from two sets.
A×B = {(a,b) : a∈A, b∈B}, |A×B| = |A|×|B|
Example: A={1,2}, B={x,y} → A×B has 4 ordered pairs.
CAT Hack: Cartesian-product pairs are ordered: (a,b) and (b,a) are different elements unless a=b.
15Exactly Two of Three Sets
Isolate the regions belonging to exactly two sets, not all three.
exactly two = [n(A∩B)+n(B∩C)+n(A∩C)] − 3·n(A∩B∩C)
Example: pairwise sums to 33, triple=5 → 33−15 = 18 people in exactly two sets.
CAT Favourite: Exactly-two and at-least-two counting is a genuine CAT favourite in Venn-diagram word problems — memorise both formulas.
CAT exam shortcuts, traps & revision
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CAT Exam Shortcuts
- Power set size = 2ⁿ, always including the empty set and the full set
- n(A∪B) = n(A)+n(B)−n(A∩B); rearrange to solve for any missing term
- Venn diagram with 3 sets: fill the innermost (triple) region first, then work outward
- De Morgan's: flip ∪⇔∩ AND complement every individual set
- only A = n(A)−n(A∩B); exactly two = Σpairwise−3·triple; at least two = Σpairwise−2·triple
- |A×B| = |A|×|B|, and every pair is ordered
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Most Common CAT Traps
- Treating “only A” as if it were simply n(A), forgetting to subtract the overlap.
- Mixing up A−B with B−A — set difference is not symmetric.
- Forgetting the final +n(A∩B∩C) term when finding a three-set union.
- Treating a Cartesian-product pair as unordered, so (a,b) and (b,a) get confused.
- Applying De Morgan's by flipping the operation but forgetting to complement each set too.
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30-Second Revision Box
- n(A∪B) = n(A)+n(B)−n(A∩B)
- n(A∪B∪C) = A−pairwise+triple
- only A = n(A)−n(A∩B); exactly two = Σpairwise−3·triple
- De Morgan's: (A∪B)′=A′∩B′; (A∩B)′=A′∪B′
- Power set size = 2ⁿ
- |A×B| = |A||B|, ordered pairs
This topic rewards reading the Venn diagram correctly over memorising every formula in isolation — once you know which region a question is really asking for, the cardinality formula is a direct substitution. Drill this sheet until the three-set union formula and the exactly-two shortcut are reflex, then test them on full sets and track progress with the CAT score predictor. It pairs directly with the Probability cheat sheet, since both lean on the same union and intersection logic. For more guides, browse the Optima Learn blog or explore every study guide, work through the CAT exam hub, and when you want mentor-led prep, book a free CAT 2026 call.
