DILR
Cubes
A one-page CAT 2026 reference for painted cube problems, covering the basic n-cube setup, corner/edge/face/interior counts, the reconciliation check, partially painted cubes, opposite versus adjacent…
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Venn Diagrams
CAT’26 DILR CHEATSHEET
Every set theory formula and shortcut you need for CAT 2026.
Venn diagram questions look simple the moment you see two or three overlapping circles, and that is exactly why small errors slip through unnoticed. The two-set formula is easy enough, but three sets bring pairwise overlaps, a triple overlap, and a habit of double or triple counting the centre if you are not careful. This sheet gathers every standard result you actually need, from the basic union and only-one regions, through the full seven-zone breakdown of a three-set diagram, to the maximum and minimum bounds an overlap can take and the subset count that closes out most set-theory basics. Every box carries one worked example with real numbers, so you drill the region-by-region logic instead of just memorising a formula.
Every Venn diagram rule you need
1Two-Set Union Formula
The total in either of two overlapping sets, counted once each.
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Example. 40 play cricket, 30 play football, 15 play both, so the union is 40 + 30 − 15 = 55 students.
CAT Hack. Adding both sets double counts the overlap, so subtracting it once fixes the count.
2Two-Set Only-One Regions
The part of each set that does not touch the other set at all.
Only A = n(A) − n(A ∩ B), Only B = n(B) − n(A ∩ B)
Example. With 40 cricket, 30 football and 15 playing both, only cricket is 40 − 15 = 25 students.
CAT Insight. The overlap always belongs to both totals, so subtract it once from each set to isolate its own zone.
3Neither Set (Two Sets)
People or items outside both sets entirely, needing a known total.
Neither = Total − n(A ∪ B)
Example. Out of 100 students with 55 in the cricket or football union, neither is 100 − 55 = 45 students.
Common Mistake. Neither uses the grand total, not just the two set sizes added together. Always confirm the total is given.
4Three-Set Union Formula
The total across three overlapping sets, correcting for every overlap.
n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C)
Example. With sizes 50, 40, 30 and pairwise overlaps 15, 10, 12 and all-three overlap 5, the union is 120 − 37 + 5 = 88.
CAT Favourite. This exact formula, and building the region-by-region grid from it, is one of the most tested set-theory setups on CAT.
5Exactly Two of Three Sets
People or items in precisely two of the three sets, never all three.
Exactly 2 = (sum of pairwise overlaps) − 3 × (all three overlap)
Example. With pairwise overlaps 15, 10, 12 and all three at 5, exactly two is (15+10+12) − 3(5) = 37 − 15 = 22.
Common Mistake. Each all-three member gets counted in all three pairwise overlaps, so subtract 3 times the triple overlap, not once.
6Exactly One of Three Sets
People or items in only one of the three sets, touching no other.
Exactly 1 = n(A∪B∪C) − (exactly 2) − (all three)
Example. With union 88, exactly two at 22 and all three at 5, exactly one is 88 − 22 − 5 = 61.
CAT Hack. Build the union first, then peel off the exactly-two and all-three layers to isolate exactly-one.
7At Least Two of Three Sets
People or items in two or more of the three sets combined.
At least 2 = (exactly 2) + (all three)
Example. With exactly two at 22 and all three at 5, at least two is 22 + 5 = 27 people.
CAT Insight. ‘At least two’ always means exactly-two plus all-three added together, never one without the other.
8Region-by-Region Grid (Seven Zones)
A three-circle Venn splits into seven distinct, non-overlapping zones.
3 exactly-one zones, 3 exactly-two zones, 1 all-three zone
Example. Fill the all-three zone first from the given overlap, then work outward to the exactly-two and exactly-one zones, so every zone value is filled in order.
CAT Favourite. Always fill the innermost all-three zone first. Every other zone in the diagram depends on it.
9Maximum Possible Overlap
The largest the intersection of two sets can be, given their sizes.
Max overlap of A, B = min(n(A), n(B))
Example. If 30 like tea and 45 like coffee, the overlap can be at most min(30, 45) = 30, since the tea drinkers cap it.
CAT Hack. The smaller set is always the ceiling on the overlap. It cannot exceed the smaller set's own size.
10Minimum Possible Overlap
The smallest the intersection of two sets can be, given a shared total.
Min overlap of A, B = n(A) + n(B) − Total
Example. Among 50 students, 30 like tea and 35 like coffee, so the overlap is at least 30+35−50 = 15 students.
Common Mistake. Minimum overlap can come out negative in the raw formula. Whenever that happens, the true minimum is 0.
11Set Difference
The elements in one set with the shared elements removed.
A − B = n(A) − n(A ∩ B)
Example. If 40 read fiction and 12 of them also read non-fiction, fiction-only readers are 40 − 12 = 28 people.
CAT Insight. Set difference is not symmetric. A minus B is not the same count as B minus A unless the sets are equal in size.
12De Morgan's Laws
Rules for negating a union or intersection of two sets.
not(A ∪ B) = (not A) ∩ (not B), not(A ∩ B) = (not A) ∪ (not B)
Example. ‘Not in cricket or football’ equals ‘not in cricket and not in football’, matching the neither-set count directly.
CAT Insight. De Morgan's laws are mostly a phrasing check. They confirm two wordings of a clue describe the same region.
13Subsets and Power Set Count
The total number of subsets, including the empty set, a set can have.
Number of subsets of an n-element set = 2n
Example. A set of 4 distinct elements has 24 = 16 subsets, including the empty set and the full set itself.
CAT Hack. Each element is independently in or out of a subset, which is the same logic as the ‘at least one selected’ count.
Exam shortcuts, common traps and revision
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CAT Exam Shortcuts
- n(A∪B) = n(A) + n(B) − n(A∩B)
- Three-set union subtracts every pair, then adds back the triple
- Exactly 2 = pairwise sum − 3 × all-three overlap
- Exactly 1 = union − exactly 2 − all three
- Max overlap = min of the two set sizes
- Min overlap = sum of sizes − total, floored at 0
- Fill the all-three zone first, then work outward
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Most Common CAT Traps
- Subtracting the triple overlap once instead of three times for exactly-two
- Forgetting to add the triple overlap back in the three-set union
- Using set sizes alone for ‘neither’ instead of the grand total
- Assuming minimum overlap can be negative instead of flooring it at 0
- Mixing up ‘at least two’ with ‘exactly two’
- Filling the outer zones of a Venn grid before the centre is known
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30-Second Revision Box
- Two-set union adds sizes, then subtracts the overlap once
- Three-set union adds back the triple overlap at the end
- Exactly one = union minus exactly two minus all three
- Neither needs the grand total, not just the set sizes
- Overlap ranges from max(0, sum − total) to min of the two sizes
- Build the Venn grid from the centre outward
The fastest route through a Venn question is almost always to fill the all-three zone first and work outward, since every other region in a three-set diagram depends on that centre value. Draw the diagram, label the zones, and let the numbers cascade rather than trying to solve everything from the formula in one step. When you want an honest read on where you stand, run a full mock and check your projected percentile with the Optima Learn CAT score predictor, then work through the DILR breakdowns on the blog. If you would rather build a plan with a person, book a free call with an Optima Learn mentor and shape one to your target.
