Syllogism • Venn diagrams • Statements and conclusions
Syllogism: Full-Length Notes
Topics Covered:
Venn Diagrams
Statements and Conclusions
1. What is a Syllogism?
A syllogism is a form of deductive reasoning where a conclusion is drawn from two or more given statements (called premises). Each statement is assumed to be 100% true, even if it contradicts general knowledge. Syllogism questions are common in competitive exams and test your ability to logically connect statements and derive valid conclusions1678.
Example:
All mammals are animals. (Major premise)
All elephants are mammals. (Minor premise)
Therefore, all elephants are animals. (Conclusion)7
2. Structure of a Syllogism
Major Premise: The general statement (e.g., All A are B).
Minor Premise: The specific statement (e.g., All B are C).
3. Types of Statements in Syllogism
| Type | Notation | Example |
|---|---|---|
| Universal Positive | A | All boys are handsome |
| Universal Negative | E | No girl is clever |
| Particular Positive | I | Some rats are dogs |
| Particular Negative | O | Some ships are not planes |
4. Venn Diagrams in Syllogism
Venn diagrams are the most reliable tool to visualize and solve syllogism problems. Each statement is represented as a circle or an overlap of circles, depending on the relationship.
How to Draw Venn Diagrams:
All A are B: Draw circle A completely inside circle B.
No A are B: Draw two non-overlapping circles.
Some A are B: Shade the overlapping area of circles A and B.
Some A are not B: Part of circle A does not overlap with circle B168.
Using Venn Diagrams:
Draw all possible diagrams for the given statements.
A conclusion is definitely true only if it holds in all possible diagrams.
If a conclusion is true in only some diagrams, it is considered not definite168.
5. Statements and Conclusions
You are given a set of statements (premises) and asked which conclusions logically follow.
Rules for Drawing Conclusions:
Definite Conclusion: Must be true in every possible Venn diagram.
Possible Conclusion: True in at least one possible diagram.
Complementary Pairs: Either-or cases arise only with specific pairs, like “Some A are B” and “No A are B”2.
Examples:
Example 1:
Statements:
All actors are right-handed.
All right-handed are artists.
Conclusion:
Some artists are actors.
Analysis:
Draw Venn diagrams: Actors ⊆ Right-handed ⊆ Artists.
Conclusion “Some artists are actors” is definitely true because all actors are artists1.
Example 2:
Statements:
All flowers are candles.
All lanterns are candles.
Conclusions:
Some flowers are lanterns.
Some candles are lanterns.
Analysis:
Draw all possible diagrams.
“Some flowers are lanterns” is not definite (could be true or false).
“Some candles are lanterns” is definitely true because all lanterns are candles5.
Example 3:
Statements:
All prisoners are men.
No man is educated.
Conclusions:
All prisoners are uneducated.
Some men are prisoners.
Analysis:
“All prisoners are men” and “No man is educated” together mean all prisoners are not educated, so both conclusions are true5.
6. Common Syllogism Patterns and Their Venn Diagrams
| Statement 1 | Statement 2 | Definite Conclusion |
|---|---|---|
| All A are B | All B are C | All A are C |
| All A are B | Some B are C | Some A are C (Possible) |
| Some A are B | All B are C | Some A are C (Possible) |
| No A is B | All B are C | No A is C |
| Some A are not B | All B are C | Some A are not C (Possible) |
7. Key Tips and Tricks
If all statements are positive, all negative conclusions are definitely false2.
“Only a few” means both “some A are B” and “some A are not B” are definitely true2.
For “either-or” conclusions, both must be possible but not definite, and must form a complementary pair (“Some + No” or “All + Some not”)2.
If a conclusion is true in any one possible diagram, it is a possible conclusion28.
If a conclusion is false in any one possible diagram, it is not definite8.
8. Practice Example
Statements:
All pens are books.
Some books are papers.
Conclusions:
Some pens are papers.
All books are pens.
Solution:
Draw Venn diagrams: Pens ⊆ Books, Papers overlap with Books.
1: “Some pens are papers” is possible but not definite.
2: “All books are pens” is false; only all pens are books.
9. Solved Example with Venn Diagram
Statements:
All cats are babies.
All babies are young.
Conclusion:
All cats are young.
Venn Diagram:
Cats ⊆ Babies ⊆ Young
So, all cats are young (definitely true)7.
10. Syllogism in Competitive Exams
Syllogism is a regular feature in exams like Bank PO, SSC, CAT, and other government exams. Mastery of Venn diagrams and logical deduction is essential for accuracy and speed16.
11. Summary Table: Syllogism Statement Types
| Statement Type | Venn Diagram Representation |
|---|---|
| All A are B | Circle A inside Circle B |
| No A is B | Separate, non-overlapping circles |
| Some A are B | Overlapping area between A and B |
| Some A are not B | Part of A outside B |
12. Final Tips
Always draw all possible Venn diagrams.
Remember: Definite conclusions must be true in every diagram.
Use the rules for complementary pairs and possible conclusions.
Practice with a variety of statement types and conclusion patterns.
References:
1: BYJU'S
2: Testbook
5: CATKing
6: Hitbullseye
7: LitCharts
8: Testbook PDF
Syllogism Practice Questions with Explanations
1.
Statements: All apples are fruits. All fruits are sweet.
Conclusions:
I. All apples are sweet.
II. Some sweet things are apples.
Answer: Both follow.
Explanation: Apples ⊆ Fruits ⊆ Sweet. So, all apples are sweet and some sweet things (specifically apples) exist.
2.
Statements: All pens are books. Some books are papers.
Conclusions:
I. Some pens are papers.
II. Some papers are books.
Answer: Only II follows.
Explanation: Some books are papers (given), but pens could be in the part of books that are not papers.
3.
Statements: Some cats are dogs. All dogs are rats.
Conclusions:
I. Some cats are rats.
II. All rats are dogs.
Answer: Only I follows.
Explanation: Cats ∩ Dogs ≠ ∅, Dogs ⊆ Rats → Cats ∩ Rats ≠ ∅. But not all rats are dogs.
4.
Statements: All roses are flowers. Some flowers are red.
Conclusions:
I. Some roses are red.
II. All flowers are roses.
Answer: Neither follows.
Explanation: Roses ⊆ Flowers, but red flowers may not include roses.
5.
Statements: No table is chair. All chairs are beds.
Conclusions:
I. No table is a bed.
II. Some beds are not tables.
Answer: Only II follows.
Explanation: All chairs are beds, so some beds (those that are chairs) are not tables.
6.
Statements: Some birds are animals. Some animals are black.
Conclusions:
I. Some birds are black.
II. Some black are animals.
Answer: Only II follows.
Explanation: Some black are animals (direct statement); birds and black may not overlap.
7.
Statements: All A are B. No B is C.
Conclusions:
I. No A is C.
II. Some B are not C.
Answer: Both follow.
Explanation: If all A are B and no B is C, then no A is C, and some B (at least A) are not C.
8.
Statements: Some books are pens. Some pens are pencils.
Conclusions:
I. Some books are pencils.
II. Some pencils are pens.
Answer: Only II follows.
Explanation: Some pencils are pens (direct), but books and pencils may not overlap.
9.
Statements: All boys are students. Some students are girls.
Conclusions:
I. Some boys are girls.
II. All girls are students.
Answer: Only II follows.
Explanation: All girls are students (given), but boys and girls may not overlap.
10.
Statements: No mango is apple. All apples are fruits.
Conclusions:
I. No mango is fruit.
II. Some fruits are apples.
Answer: Only II follows.
Explanation: Apples ⊆ Fruits, so some fruits are apples.
11.
Statements: Some plums are peaches. All peaches are apples. Some apples are mangoes.
Conclusions:
I. Some mangoes are peaches.
II. Some apples are plums.
Answer: Only II follows7.
Explanation: Some apples are plums (since peaches ⊆ apples and some plums are peaches).
12.
Statements: All flowers are beautiful. Vaidehi is beautiful.
Conclusions:
I. Vaidehi is a flower.
II. Some beautiful are flowers.
Answer: Only II follows2.
Explanation: Vaidehi may or may not be a flower, but some beautiful are flowers (since all flowers are beautiful).
13.
Statements: All pens are pencils. No pencil is eraser.
Conclusions:
I. No pen is eraser.
II. Some pencils are pens.
Answer: Both follow.
Explanation: Pens ⊆ Pencils, so some pencils are pens; pens ⊆ pencils, and no pencil is eraser, so no pen is eraser.
14.
Statements: Some dogs are cats. Some cats are rats.
Conclusions:
I. Some dogs are rats.
II. Some rats are cats.
Answer: Only II follows.
Explanation: Some rats are cats (direct), but dogs and rats may not overlap.
15.
Statements: All tigers are jungles. All elephants are jungles. All jungles are trees.
Conclusions:
I. All tigers are trees.
II. No tigers are elephants.
III. Some tigers are elephants.
Answer: Only I follows1.
Explanation: Tigers ⊆ Jungles ⊆ Trees; nothing about tigers and elephants' overlap.
16.
Statements: Some Black is White. Some Pink is Grey. Some Grey is White.
Conclusions:
I. Some Grey is not Black.
II. At least some Black can be Grey.
Answer: Only II follows2.
Explanation: Possibility for Black and Grey to overlap exists.
17.
Statements: All Z are D. No Y is Z.
Conclusions:
I. Some D are not Y.
II. Some Z are not Y.
III. Some D are Z.
Answer: All follow2.
Explanation: Z ⊆ D, no Y is Z, so some D (the Z part) are not Y, etc.
18.
Statements: Some schools are houses. Some colleges are schools.
Conclusions:
I. Some colleges are houses.
II. Some colleges are not houses.
Answer: Only II follows2.
Explanation: Possible for colleges and houses to not overlap.
19.
Statements: No fingers are legs. Mostly legs are hands. Only a few hands are hairs.
Conclusions:
I. Some legs are not hairs.
II. All hands being fingers is a possibility.
Answer: Only II follows6.
Explanation: All hands can be fingers, but some legs not being hairs is not definite.
20.
Statements: All birds are animals. Some animals are mammals.
Conclusions:
I. Some birds are mammals.
II. All mammals are animals.
Answer: Only II follows.
Explanation: All mammals are animals (direct), but birds and mammals may not overlap.
21.
Statements: All pens are books. Some pencils are books.
Conclusions:
I. Some pencils are pens.
II. All pens are pencils.
Answer: Neither follows.
Explanation: Pencils and pens may not overlap.
22.
Statements: Some oranges are apples. Some apples are bananas.
Conclusions:
I. Some oranges are bananas.
II. Some bananas are apples.
Answer: Only II follows.
Explanation: Some bananas are apples (direct); oranges and bananas may not overlap.
23.
Statements: All roses are flowers. All flowers are plants.
Conclusions:
I. All roses are plants.
II. Some plants are roses.
Answer: Both follow.
Explanation: Roses ⊆ Flowers ⊆ Plants.
24.
Statements: No boy is girl. All girls are students.
Conclusions:
I. No boy is student.
II. Some students are girls.
Answer: Only II follows.
Explanation: Some students are girls (direct); boys and students may overlap.
25.
Statements: Some pens are pencils. Some pencils are erasers.
Conclusions:
I. Some pens are erasers.
II. Some erasers are pencils.
Answer: Only II follows.
Explanation: Some erasers are pencils (direct).
26.
Statements: All A are B. All B are C.
Conclusions:
I. All A are C.
II. Some C are A.
Answer: Both follow.
Explanation: A ⊆ B ⊆ C.
27.
Statements: Some A are B. All B are C.
Conclusions:
I. Some A are C.
II. All C are B.
Answer: Only I follows.
Explanation: Some A ⊆ B ⊆ C, so some A are C.
28.
Statements: All A are B. Some B are C.
Conclusions:
I. Some A are C.
II. All C are B.
Answer: Neither follows.
Explanation: A ⊆ B, some B ⊆ C, but A and C may not overlap.
29.
Statements: No A is B. All B are C.
Conclusions:
I. No A is C.
II. Some C are not A.
Answer: Only II follows.
Explanation: All B ⊆ C, A and B disjoint, but A and C may overlap outside B.
30.
Statements: Some A are B. Some B are C.
Conclusions:
I. Some A are C.
II. All C are A.
Answer: Neither follows.
Explanation: A and C may not overlap.
Comments
Post a Comment