Syllogisms & Statements

Snapshot

• Syllogisms give you two or three statements about groups ("All A are B", "Some B are C", "No C is D") and ask which conclusions must follow. Real-world knowledge is irrelevant: treat every statement as true, even if it is absurd.
• CUET asks five things: which conclusion follows, "either I or II", "neither follows", possibility-type conclusions, and which Venn diagram fits three given classes.
• The idea that cracks it: a conclusion follows only if it is true in every diagram you can draw. If you can draw the statements so the conclusion fails, it does not follow.
• Marking is +5 / −1. Draw circles for every question, however easy it looks.

Part 1 — The four statement types and their pictures

Written and reviewed by Devendra Dubey, Mathematics & Commerce faculty at UniDrill (M.Sc. Mathematics, Osmania University · 10 years teaching). Prepared for the CUET UG General Test.

Every syllogism statement is one of four kinds, and each has a standard Venn picture. Learn the pictures, not the words.

Two conventions make the whole topic precise. First, "Some" means at least one and possibly all, so "Some A are B" does not say that some A are not B. Second, every group named in a statement is assumed to have at least one member, so "All A are B" guarantees there are some A and they are all B. These are the conventions the exam uses.

The pictures above are the least you can say. "Some A are B" is drawn as overlap, but the circles could also be drawn one inside the other or even identical. That is exactly why the test for a conclusion is to see whether it survives every legal redrawing.

Part 2 — What each statement lets you say immediately

Before combining statements, know what a single statement already gives you. This table is the engine behind most "which conclusion follows" questions.

Notice the two most-forgotten facts: "All A are B" does give "Some B are A", and "Some A are not B" does not give "Some B are not A". A small circle drawn inside a large one shows both at a glance.

Part 3 — Combining two statements

When two statements share a middle term ($A$ with $B$, then $B$ with $C$), the pair gives a definite relation between $A$ and $C$ only in certain combinations. The table assumes the shared term $B$ appears in both statements as written below. If it is reversed, convert the statement first using Part 2 if conversion is allowed.

A good memory aid is the "weakest link": a conclusion is never stronger than the weakest statement behind it. If a "Some" is anywhere in the chain, the conclusion can say at most "Some". If a "No" is in the chain, the conclusion is negative.

Part 4 — The method, step by step

For three statements, join the first two to get a single relation, then join that result with the third. If the first two give nothing definite, a conclusion linking the end terms rarely follows. For any set, follow these steps.

1. Read the statements as true even when they are silly ("All stones are trees" is true for this question).
2. Draw the statements using the pictures in Part 1. Keep every circle on one diagram.
3. Draw the conclusion's opposite: try to arrange the circles so that the conclusion becomes false while the statements stay true.
4. If you can break it, it does not follow. If you cannot break it however you try, it follows.
5. Check the converse and complement pairs (Part 5) before marking "neither".
6. Pick the answer option that matches the set of conclusions that follow.

Part 5 — Either-or, neither, and possibility

Either I or II. Pick this when neither conclusion follows on its own, but one of them must be true in every diagram. This happens only when the two conclusions are about the same two terms and are opposites in a precise sense:

The last row is the trap: "All" and "No" together are called contraries. Both can be false at once (some A are B and some are not), so the answer is "neither".

Neither follows. The default when no conclusion is certain and no complementary pair applies.

Possibility conclusions. Words like "can be", "may be" or "is a possibility" make the conclusion weaker: it follows if at least one valid diagram makes it true, and fails only when the statements directly forbid it. "Some A can be C" fails if "No A is C" is forced, and passes in almost every other case.

Part 6 — The full menu of question types

Part 7 — Picking the right Venn diagram for three classes

This question type gives three words (for example Animals, Cows, Dogs) and asks which diagram represents them. Ask the same question for each pair: is one always inside the other, do they sometimes overlap, or are they always separate?

Animals, Cows, Dogs: every cow and every dog is an animal, and no cow is a dog, so two separate small circles sit inside one large circle. Males, Fathers, Doctors: every father is male, so Fathers is inside Males. A doctor may be male or female, and may or may not be a father, so the Doctors circle cuts across Males and across Fathers. Three standard shapes cover most questions: all three nested (for example Delhi, India, Asia), two separate circles inside one (Animals, Cows, Dogs) and three overlapping circles (Boys, Students, Girls: Boys and Girls separate, each overlapping Students).

Part 8 — Worked examples

Example 1. A chain of "All". Statements: All pens are books. All books are red. Conclusions: I. All pens are red. II. Some red things are pens. III. All red things are pens.

Pens sit inside books, which sit inside red. I follows. "All pens are red" gives "Some red things are pens", so II follows. III reverses an "All", so it does not follow. Answer: I and II.

Example 2. "Some" followed by "All". Statements: Some cats are dogs. All dogs are animals. Conclusions: I. Some cats are animals. II. Some animals are cats. III. All cats are animals.

The cats that are dogs are inside the animals, so I follows, and by reversing a "Some", II follows. Some cats might not be dogs, so III does not follow. Answer: I and II.

Example 3. Two "Some". Statements: Some A are B. Some B are C. Conclusions: I. Some A are C. II. Some C are A.

Draw A and C as two separate circles, each overlapping B. The statements hold and A and C do not touch. So I and II both fail. Answer: neither follows.

Example 4. "All" then "No". Statements: All roses are flowers. No flower is red. Conclusions: I. No rose is red. II. No red thing is a rose. III. Some flowers are red.

Roses are inside flowers, and flowers are separate from red, so I follows. II is the reverse of I, so it follows. III contradicts the second statement. Answer: I and II.

Example 5. A conclusion that points the wrong way. Statements: Some books are pens. No pen is a pencil. Conclusions: I. Some books are not pencils. II. Some pencils are not books.

The books that are pens are not pencils, so I follows. For II, the pencil circle can be drawn entirely inside books, so some pencils are books and none need be outside. II fails. Answer: only I.

Example 6. The contraries trap. Statement: Some A are B. Conclusions: I. All A are B. II. No A is B.

II is directly contradicted. I is possible but not certain, since the circle A can stick out of B. Neither follows, and the pair is "All" and "No", which is not an either-or pair. Answer: neither follows.

Example 7. A contradictory pair. Statement: Some doctors are teachers. Conclusions: I. All doctors are teachers. II. Some doctors are not teachers.

Neither is certain: the doctors could all be teachers, or some might not be. They are a complementary pair ("All" and "Some not") that cannot both be false. Answer: either I or II follows.

Example 8. Either-or after a chain. Statements: All X are Y. Some Y are Z. Conclusions: I. Some X are Z. II. No X is Z.

Draw X inside Y, and Z overlapping Y. Z can overlap the X part of Y or only the rest of Y, so neither I nor II is certain. They are a "Some" and "No" pair on the same terms. Answer: either I or II follows.

Example 9. Three statements. Statements: All A are B. Some B are C. No C is D. Conclusions: I. Some B are not D. II. Some A are not D. III. No A is D.

Some B are C, and no C is D, so those B are not D. I follows. For II and III, A might lie wholly outside C or even overlap D, so neither is certain. Answer: only I.

Example 10. A longer chain. Statements: All A are B. All B are C. Some C are D. Conclusions: I. Some C are A. II. Some A are D. III. Some D are C.

All A are B and all B are C, so all A are C, and by reversing, some C are A. I follows. III reverses a "Some", so it follows. For II, D may sit in C outside A. Answer: I and III.

Example 11. Possibility. Statements: Some A are B. All B are C. Conclusions: I. Some A are C. II. All A can be C. III. No A is C.

The A that are B are inside C, so I follows. The remaining A (if any) could also be inside C, so II is a valid possibility. III is forbidden. Answer: I and II.

Example 12. Which diagram? Which relationship shows Boys, Students, Girls?

Boys and Girls cannot overlap each other, but a boy may be a student and a girl may be a student, so Students overlaps both. Answer: two separate circles, each overlapping a third circle.

Example 13. A false statement. If the statement "All birds are fliers" is false, which statement must be true?

If "All A are B" is false, then at least one A is not B. The answer is Some birds are not fliers.

Part 9 — Common traps

Part 10 — Rapid-fire practice with answers

In each row, decide whether the stated conclusion follows.

Part 11 — A timed-practice routine

In the exam, give a two-statement set about a minute. If you get stuck, draw the weakest possible overlap (the two circles touching only at the "Some" part) and test each conclusion against it. Do not search for the answer by feel.

Part 12 — Where this connects

The same "true in every case" discipline underlies Classification, where you decide what a group has in common, and Analogy, where you identify the relationship between two classes. Chain-building is shared with Ranking & Arrangement. Reading statements precisely, without bringing in outside facts, also helps in Blood Relations.

One-line revision: draw every statement as circles, accept all statements as true, a conclusion follows only if it survives every drawing, two "Some" or two "No" give nothing, and only complementary pairs make "either I or II".