DILR
Distribution
A one-page CAT 2026 reference for distribution problems, covering stars and bars with and without empty groups, distinct items into distinct groups, non-negative and positive integer solutions,…
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Binary Logics
CAT’26 DILR CHEATSHEET
Every truth, liar and if-then rule you need for CAT 2026 binary logic sets.
Binary logic puzzles strip a DILR set down to its cleanest possible form, since every statement in front of you is either fully True or fully False, with nothing in between to guess at. That simplicity is exactly what makes them fast once you know the moves and slow once you do not, because the whole puzzle usually turns on one clean piece of structure, a contradictory pair, a lone liar, an if-then clue that only fails in one case. This sheet lays out the setups you actually meet, from the basic true or false rule, through truth-tellers and liars, if-then statements and their contrapositives, to the case elimination habit that closes most of these puzzles in a few clean steps. Every box carries one worked mini example, so you drill the reasoning and not just the definition.
Every binary logic rule you need
1Binary Truth Basics
Every statement in these puzzles is strictly True or False, nothing between.
A statement is T or F, never partly true
Example. If ‘X stole the wallet’ is False, then X did not steal it, full stop, with no partial or maybe reading allowed.
CAT Hack. Treat every statement as a switch, T or F, and test both settings before trusting your gut read.
2Contradictory Statements
Two statements that cannot both be true and cannot both be false.
If P and Q contradict, exactly one of P, Q is True
Example. ‘A is the tallest’ and ‘A is not the tallest’ contradict, so exactly one of the two is True, always.
CAT Favourite. Spotting a contradictory pair instantly tells you one is true and one is false, without checking any other clue.
3Complementary Statements
Two statements that together must cover every possible case.
If P and Q are complementary, at least one of P, Q is True
Example. ‘The score is even’ and ‘The score is odd’ are complementary, so at least one holds, though unlike contradiction both being true is impossible only if they also conflict.
CAT Insight. Complementary is not the same as contradictory. Check whether both being false is actually possible before assuming a pair is contradictory.
4Exactly One Statement True
Among several statements, precisely one is True and the rest are False.
Assume each statement True in turn, keep the case with no clash
Example. Among 3 claims, if assuming claim 2 is True makes claims 1 and 3 correctly False with no clash, claim 2 is the one true statement.
CAT Hack. Run every statement as the ‘True one’ in turn and discard any case where a second statement also turns out True.
5Exactly One Statement False
Among several statements, precisely one is False and the rest are True.
Assume each statement False in turn, keep the case with no clash
Example. Among 4 claims, testing claim 3 as the False one and checking the other three still hold True together confirms claim 3 is the one false statement.
CAT Hack. This is the mirror of exactly-one-true. Flip your assumption and run the same case check.
6Truth-Teller and Liar Types
Some people always tell the truth, others always lie, with no in-between.
Truth-teller's statements are all True, liar's are all False
Example. If B is a liar and B says ‘C is honest’, then C being honest is False, so C is actually a liar too.
Common Mistake. A liar does not randomly lie once. Every single statement from a liar must be False, so check all of them, not just one.
7Testing a Person as the Truth-Teller
Assume one person always tells the truth and trace the consequences.
Assume X truthful, mark all of X's statements True, then check for clashes
Example. Assume D is truthful. If D says ‘E is lying’, then E's statements must all be False, and if that breaks another clue the assumption on D fails.
CAT Hack. Start with whichever person's statement gives the most immediate consequences, since that shuts down bad branches fastest.
8Chain of Implications (If-Then)
An if-then statement is False only when the first part is True and the second is False.
‘If P then Q’ is False only when P is True and Q is False
Example. ‘If it rains, the match is cancelled’ is broken only by rain with no cancellation. No rain at all keeps the statement True regardless of the match.
Common Mistake. An if-then statement is not automatically False just because the first part did not happen. Check the one specific breaking case only.
9Contrapositive Rule
An if-then statement and its reversed, negated form always share the same truth value.
‘If P then Q’ has the same truth value as ‘If not Q then not P’
Example. ‘If X is a manager, X attends the meeting’ is logically identical to ‘If X did not attend, X is not a manager’, giving a second useful clue for free.
CAT Favourite. Rewriting a clue as its contrapositive often reveals a fact the original phrasing hid, and it costs nothing to write out.
10Binary Attribute Grid
Track yes or no answers across people and properties in a grid.
Rows are people, columns are properties, each cell is Yes or No
Example. Mark Yes for ‘plays chess’ against each person's row, and once a row has enough Yes marks to satisfy a total clue, the remaining cells lock to No.
CAT Favourite. The grid does for binary logic what the seating chart does for arrangements. Build it before guessing.
11At Least One True Among a Set
A condition guaranteeing that not every statement in a set is False.
At least one True rules out the all-False case only
Example. If at least one of 3 alibis is True, the single scenario where all 3 are False is the only one you can eliminate outright, leaving several other combinations open.
Common Mistake. ‘At least one true’ is a weak clue on its own. It removes one case, not most of them, so pair it with other conditions.
12Self-Referential and Paradox Statements
A statement that talks about its own truth value needs careful handling.
Test both T and F assignments for internal consistency
Example. ‘This statement is False’ breaks under both assignments, so CAT puzzles avoid true paradoxes. If one appears, check for a misread condition instead.
CAT Insight. A genuine unsolvable paradox almost never appears in a real CAT set. Recheck the wording before assuming one exists.
13Case Elimination by Clash
Discard an assumed case the moment it produces two conflicting facts.
One clash anywhere in the case is enough to reject the whole case
Example. While testing ‘F is truthful’, if that forces both ‘G is present’ and ‘G is absent’ to hold, the case is rejected immediately, no further checking needed.
CAT Hack. Stop checking a case the instant you spot one clash. There is no need to work through the remaining clues.
Exam shortcuts, common traps and revision
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CAT Exam Shortcuts
- Every statement is strictly True or False, never partial
- A contradictory pair always has exactly one True
- Exactly-one-true or exactly-one-false, test each statement in turn
- A liar's every statement is False, not just one of them
- An if-then clue is False in only one specific case
- Contrapositive gives a free extra clue from the same fact
- One clash anywhere rejects the whole case immediately
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Most Common CAT Traps
- Confusing a contradictory pair with a complementary pair
- Believing a liar told the truth just once by mistake
- Marking an if-then clue False when the first part simply did not happen
- Stopping after finding one consistent case without checking for a second one
- Treating ‘at least one true’ as if it fixes most of the answer
- Forgetting to test every statement in turn for an exactly-one condition
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30-Second Revision Box
- Statements are strictly True or False, no middle ground
- Contradiction forces exactly one True, complementary is weaker
- Truth-tellers and liars are all-or-nothing across every statement
- If-then fails only when the first part is True and the second False
- Write the contrapositive for a free second clue
- Drop a case the moment any clash appears
Binary logic rewards a systematic assumption over a clever guess, so pick the statement with the biggest consequences, assume a truth value, and follow it until it either holds together or clashes. The moment a clash appears, drop the case and move to the next one without a second thought. 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.
