Logical inference — SSC CGL Reasoning Notes

Concepts and rules

A syllogism gives statements about classes and asks which conclusions must be true. Accept the statements as true even if they sound absurd, and never use outside knowledge.

The 4 statement types: All A are B (A circle fully inside B); No A is B (A and B do not touch); Some A are B (at least one A is a B, possibly all); Some A are not B (at least one A lies outside B).

'Some' means at least one and does not exclude 'all'. So from 'Some A are B' you cannot conclude 'Some A are not B'.

Every class mentioned is taken to have at least one member, so 'All A are B' also gives 'Some A are B' and 'Some B are A'.

Valid conversions: All A are B → Some B are A (not All B are A). No A is B → No B is A. Some A are B → Some B are A. 'Some A are not B' cannot be converted at all.

Chaining through the middle term B: All A are B + All B are C → All A are C. All A are B + No B is C → No A is C. Some A are B + All B are C → Some A are C. Some A are B + No B is C → Some A are not C.

No definite link: All A are B + Some B are C, and Some A are B + Some B are C, give no definite relation between A and C, because the C part of B may or may not touch A.

Either-or: when neither of 2 conclusions follows but they are about the same 2 classes and one of them must be true (for example 'Some A are C' and 'No A is C'), the answer is 'either I or II follows'.

Possibility conclusions ('All A being C is a possibility') follow if at least one valid diagram makes them true, that is, if no statement forbids them.

Method and shortcuts

Step 1: Draw the basic diagram using the least overlap the statements allow: 'Some' as a small overlap, 'All' as a circle inside, 'No' as separate circles.

Step 2: For each conclusion, try to draw a second valid diagram in which it is false. If you succeed, it does not follow; if every attempt breaks a statement, it follows.

Step 3: Before rejecting 2 failed conclusions, check whether they form an either-or pair on the same 2 classes.

Shortcut: a conclusion with 'All' needs a chain of 'All' statements; any 'Some' in the chain makes 'All' impossible to conclude.

Shortcut: a negative conclusion (No, Some not) needs at least one negative statement; 2 positive statements never give a negative conclusion.

Example 1: a chain of All

Question: Statements: All pens are books. All books are tables. Conclusions: I. All pens are tables. II. Some tables are pens.

Solution: Pens sit inside books, which sit inside tables, so every pen is a table (I). Pens exist and are tables, so some tables are pens (II).

Answer: Both I and II follow.

Example 2: Some with No

Question: Statements: Some cats are dogs. No dog is a rat. Conclusions: I. Some cats are not rats. II. No cat is a rat.

Solution: The cats that are dogs cannot be rats, so I follows. But the other cats are free: a diagram where some non-dog cats are rats breaks no statement, so II does not follow.

Answer: Only I follows.

Example 3: no definite link

Question: Statements: All roses are flowers. Some flowers are red. Conclusions: I. Some roses are red. II. Some red things are flowers.

Solution: The red part of flowers may lie outside the roses circle, so I does not follow. II is the conversion of 'Some flowers are red', so it follows.

Answer: Only II follows.

Example 4: All with No

Question: Statements: No apple is a mango. All mangoes are fruits. Conclusions: I. Some fruits are not apples. II. No apple is a fruit. (a) Only I follows (b) Only II follows (c) Both follow (d) Neither follows

Solution: The mangoes are fruits and are not apples, so some fruits (the mangoes) are not apples: I follows. Apples may still overlap the rest of the fruits circle, so II does not follow.

Note: 'No apple is a fruit' is a common trap; the 'No' statement is about mangoes, not about fruits.

Answer: (a) Only I follows.

Example 5: an either-or pair

Question: Statements: All cups are plates. Some plates are bowls. Conclusions: I. Some cups are bowls. II. No cup is a bowl. (a) Only I follows (b) Only II follows (c) Either I or II follows (d) Neither I nor II follows

Solution: The bowls may touch the cups circle or stay away from it, so neither I nor II is certain. Both are about cups and bowls, and in every diagram exactly one of them is true.

Answer: (c) Either I or II follows.

Example 6: three statements, four conclusions

Question: Statements: Some pencils are erasers. All erasers are sharpeners. No sharpener is a scale. Conclusions: I. Some pencils are sharpeners. II. Some pencils are not scales. III. No eraser is a scale. IV. Some scales are pencils. (a) Only I and III (b) Only I, II and III (c) Only II and IV (d) All four

Solution: Some + All gives I: the pencils that are erasers are sharpeners. All + No gives III: erasers lie inside sharpeners, which never touch scales. Those same pencils are therefore not scales, so II follows.

IV: other pencils, outside the sharpeners, may or may not be scales, so IV is not certain.

Answer: (b) Only I, II and III.

Common mistakes

Converting 'All A are B' into 'All B are A'. The correct conversion is 'Some B are A'.

Reading 'Some A are B' as 'Some A are not B'. Some allows all.

Linking A and C through a middle term that is only 'Some' on both sides. Some + Some gives nothing.

Marking 'either-or' for any 2 doubtful conclusions. The pair must be about the same 2 classes and one must always hold.

Using real-world facts. If the statements say all fish fly, accept it for the question.

Practice questions

Q1. Statements: All doctors are graduates. No graduate is illiterate. Conclusions: I. No doctor is illiterate. II. Some graduates are doctors. (a) Only I (b) Only II (c) Both I and II (d) Neither I nor II

Q2. Statements: Some books are pens. Some pens are pencils. Conclusions: I. Some books are pencils. II. No book is a pencil. (a) Only I (b) Either I or II (c) Neither I nor II (d) Only II

Q3. Statements: Some trees are tall. No tall thing is a shrub. Conclusions: I. Some trees are not shrubs. II. Some shrubs are trees. III. All trees being shrubs is a possibility. (a) Only I (b) Only I and III (c) Only II (d) Only I and II

Answers

Q1. (c) All + No gives I. All doctors are graduates and doctors exist, so some graduates are doctors (II).

Q2. (b) Some + Some gives no definite link, so neither is certain; they cover books and pencils and one of them must be true.

Q3. (a) The tall trees are not shrubs, so I follows and III is impossible. Shrubs may avoid the trees completely, so II does not follow.

Quick revision

Follows = true in every valid diagram; possibility = true in at least one.

All+All = All; All+No = No; Some+All = Some; Some+No = Some not; Some+Some = nothing.

All A are B converts only to Some B are A; Some not never converts.

Either-or: same 2 classes, neither certain, one always true.

Concepts and rules

Any 2 classes stand in exactly one of 3 relations: one is inside the other (all members of one belong to the other), they overlap (some members are common, some are not), or they are separate (nothing in common).

Diagrams for 3 classes, described in words: 3 concentric circles (A inside B inside C); 2 separate circles inside a third (2 kinds of one thing); 2 overlapping circles inside a third; 3 mutually overlapping circles; one circle inside another with a third separate; one circle inside another with a third overlapping both; 3 separate circles.

Choose the diagram from the 3 pairwise relations, not from the first word. For example Delhi, India, Asia: Delhi is inside India and India is inside Asia, so the diagram is concentric.

Roles such as doctor, woman, singer, parent overlap with each other, because one person can hold several roles. Kinds of one thing (dog and cat as animals) are separate inside the bigger class.

Two sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Only A = n(A) − n(A ∩ B). Neither = total − n(A ∪ B).

Three sets: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C). The triple part is subtracted 3 times in the pair terms, so it is added back once.

Only A (among 3 sets) = n(A) − n(A ∩ B) − n(A ∩ C) + n(A ∩ B ∩ C). Exactly 2 of the sets = (sum of pair overlaps) − 3 × (triple overlap).

If everyone belongs to at least one set, n(A ∩ B) = n(A) + n(B) − total.

Method and shortcuts

Step 1 for diagram choice: write the 3 pairs (A-B, B-C, A-C) and label each inside, overlap or separate; then match the pattern to the diagram.

Step 2 for counts: fill the Venn regions from the centre outward: triple part first, then 'exactly two' parts, then 'only' parts, then 'none'.

Step 3: check that all regions add to the total; a negative region means a misread number.

Shortcut: 'pair overlap' figures in questions usually include the triple part; subtract the triple part to get 'exactly those two'. If the question says 'only tea and coffee', the triple part is already excluded.

Example 1: kinds inside a class

Question: Which diagram shows Mammals, Dogs and Cats?

Solution: Dogs are inside mammals, cats are inside mammals, and no dog is a cat. Pattern: inside, inside, separate.

Answer: 2 separate circles inside a bigger circle.

Example 2: a chain of inclusion

Question: Which diagram shows Delhi, India and Asia?

Solution: Delhi is inside India and India is inside Asia, so Delhi is also inside Asia.

Answer: 3 concentric circles.

Example 3: two-set survey

Question: Of 60 students, 35 like tea, 30 like coffee and 10 like both. How many like neither?

Solution: n(tea ∪ coffee) = 35 + 30 − 10 = 55. Neither = 60 − 55 = 5. Regions: only tea 25, only coffee 20, both 10, neither 5, total 60.

Answer: 5.

Example 4: roles overlap

Question: Which diagram best represents Doctors, Women and Singers? (a) 3 concentric circles (b) 3 mutually overlapping circles (c) 2 separate circles inside a third (d) 3 separate circles

Solution: Some doctors are women and some are not; some women are singers and some are not; some doctors sing and some do not. All three pairs overlap, and a woman doctor who sings is possible.

Answer: (b) 3 mutually overlapping circles.

Example 5: three-set survey

Question: In a group of 100 people, 50 read paper A, 40 read B and 30 read C; 15 read A and B, 10 read B and C, 12 read A and C, and 5 read all three. How many read exactly one paper? (a) 56 (b) 88 (c) 66 (d) 61

Solution: Only A = 50 − 15 − 12 + 5 = 28. Only B = 40 − 15 − 10 + 5 = 20. Only C = 30 − 10 − 12 + 5 = 13. Exactly one = 28 + 20 + 13 = 61.

Check: exactly two = (15 + 10 + 12) − 3 × 5 = 22, all three = 5, so readers = 61 + 22 + 5 = 88 and non-readers = 12. Option (b) 88 is the number reading at least one.

Answer: (d) 61.

Example 6: reading a diagram with numbers

Question: In a diagram, the circle is graduates, the triangle is employed people and the rectangle is women. The regions hold: graduate only 12, employed only 9, women only 15, graduate and employed only 6, graduate and women only 8, employed and women only 5, all three 4. How many graduates are employed or women, but not both? (a) 18 (b) 10 (c) 14 (d) 22

Solution: Inside the circle, 'employed but not woman' is the region graduate and employed only = 6, and 'woman but not employed' is graduate and women only = 8. The region with all three is excluded because it is both. Total = 6 + 8 = 14.

Option (a) 18 wrongly adds the 4 who are in all three.

Answer: (c) 14.

Common mistakes

Choosing a diagram from 1 pair only. Check all three pairs; Women, Mothers, Teachers is not 3 mutually overlapping circles, because every mother is a woman.

Adding 2 sets without subtracting the overlap, which counts the common members twice.

Forgetting to add back the triple part in the 3-set formula.

Treating 'read A and B' as 'read only A and B'. Unless 'only' is written, the figure includes those who read all three.

Practice questions

Q1. Which diagram best represents Stationery, Pens and Notebooks? (a) 3 concentric circles (b) 3 mutually overlapping circles (c) 2 separate circles inside a third (d) 3 separate circles

Q2. In a group of 80 people, 50 like cricket and 45 like football, and everyone likes at least one of the two. How many like both? (a) 10 (b) 15 (c) 20 (d) 25

Q3. Which diagram best represents Women, Mothers and Teachers? (a) One circle inside another, with a third circle overlapping both (b) 3 concentric circles (c) 2 separate circles inside a third (d) 3 mutually overlapping circles

Answers

Q1. (c) Pens and notebooks are both stationery and no pen is a notebook.

Q2. (b) Both = 50 + 45 − 80 = 15.

Q3. (a) Every mother is a woman, so Mothers lies inside Women. Some teachers are women and some are not, and some teachers are mothers and some are not, so Teachers overlaps both.

Quick revision

Label every pair: inside, overlap or separate; then pick the diagram.

n(A ∪ B) = n(A) + n(B) − n(A ∩ B); for 3 sets subtract the pairs and add the triple.

Fill regions from the centre outward and check the total.