Unit 2: Syllogism, Time Sequence and Ranking - Subjective Questions

A logical Venn diagram is a diagrammatic method of representing the relationship between different classes or categories. Each class is usually shown by a circle.

The four basic categorical statements are represented as follows:

• All A are B: The circle representing A lies completely inside the circle representing B. Symbolically, .
• No A is B: The circles representing A and B are separate and do not overlap. Thus, .
• Some A are B: The circles representing A and B overlap, and an element is placed in the common region. Thus, .
• Some A are not B: A part of circle A lies outside circle B, and an element is marked in that region.

Venn diagrams help determine whether a conclusion follows definitely, follows only as a possibility, or does not follow from the given statements.

Let the classes be poets , thinkers , teachers , and pilots .

• From All poets are thinkers, place circle completely inside circle : .
• From Some thinkers are teachers, circles and must overlap: .
• From No teacher is a pilot, circles and must be disjoint: .

The diagram does not fix a definite relationship between poets and teachers because the poets may or may not lie in the part of thinkers that overlaps teachers. Similarly, no definite relationship can be established between thinkers and pilots.

Therefore:

• Some thinkers are teachers is definite.
• No teacher is a pilot is definite.
• Some poets being teachers is possible, but not definite.
• Some thinkers being pilots is also possible, provided those thinkers are not teachers.

Let engineers be , graduates be , artists be , and bankers be .

Given:

Conclusion (i): Some engineers are artists.

This does not follow. The graduates who are artists need not be engineers.

Conclusion (ii): No engineer is a banker.

This does not follow. There is no direct restriction between engineers and bankers.

Conclusion (iii): Some graduates are not bankers.

This follows. Some graduates are artists, and no artist is a banker. Therefore, those graduates who are artists are not bankers.

Hence, only conclusion (iii) follows.

A minimum possible diagram represents only those relationships that are explicitly required by the statements. It avoids assuming any additional overlap, separation, or existence.

An alternative diagram represents another valid arrangement of the same classes without violating the statements.

Their roles are:

• A conclusion is definite only when it remains true in every valid diagram.
• A conclusion is possible when it is true in at least one valid diagram.
• A conclusion is invalid when it cannot be true in any valid diagram.
• Alternative diagrams prevent unjustified assumptions about classes whose relationship is not specified.

For example, if all are and all are , classes and may overlap, remain separate, or one may contain the other. Therefore, some A are C is possible but not definite.

A possibility-based conclusion states that a particular relationship may exist rather than asserting that it must exist.

A possibility conclusion is accepted when:

• At least one valid Venn diagram can represent the proposed relationship.
• The proposed relationship does not contradict any definite statement.
• No universal negative or restrictive statement directly prevents the relationship.

It is rejected when:

• Every valid diagram contradicts the proposed relationship.
• A definite statement directly prohibits it.
• Accepting it would place an element simultaneously in mutually exclusive classes.

For example, from No A is B, the conclusion Some A being B is possible is rejected because . However, from All A are C and All B are C, some A being B is possible because no statement forces A and B to remain separate.

Let doctors be , researchers be , careless persons be , and musicians be .

Given:

(i) Some doctors are musicians: This is possible. A musician who is not careless may also be a doctor. The statement only requires some musicians, not all musicians, to be careless.

(ii) Some musicians are researchers: This is possible for the same reason. The careless musicians cannot be researchers, but other musicians may be researchers.

(iii) No doctor is careless: This is definite. Since every doctor is a researcher and no researcher is careless, .

Thus, possibilities (i) and (ii) are valid, while statement (iii) is a definite conclusion.

A definite conclusion must be true in every arrangement that satisfies the given statements. A possible conclusion needs to be true in only one valid arrangement and must not contradict the statements.

Definite conclusion example:

• Statements: All roses are flowers. No flower is a machine.
• Conclusion: No rose is a machine.
• Reason: Since and , it necessarily follows that .

Possible conclusion example:

• Statements: All pens are stationery. All pencils are stationery.
• Conclusion: Some pens being pencils is possible.
• Reason: Pens and pencils may overlap inside the stationery class, although such overlap is not compulsory.

Therefore, definite conclusions express necessity, whereas possible conclusions express logical compatibility.

Let books be , journals be , publications be , and handwritten items be .

Given:

(i) Some books being handwritten is possible: Yes. The books that are journals cannot be handwritten, but other books may be handwritten because all books are not stated to be publications.

(ii) Some books are not handwritten: This follows definitely. Some books are journals, all journals are publications, and no publication is handwritten. Hence those books that are journals are not handwritten.

(iii) No journal is handwritten: This also follows definitely because and .

Thus, (i) is a valid possibility, while (ii) and (iii) are definite conclusions.

A number test is a reasoning problem in which digits or numbers are examined according to specified rules. It tests observation, numerical arrangement, counting, and logical manipulation rather than advanced arithmetic.

Common operations include:

• Arranging digits in ascending or descending order.
• Interchanging specified digits or positions.
• Adding, subtracting, multiplying, or dividing selected digits.
• Comparing place values.
• Identifying odd, even, prime, or composite digits.
• Counting digits satisfying a condition.
• Forming the greatest or smallest possible number.
• Finding unchanged positions after rearrangement.
• Comparing adjacent digit differences.

The rules must be applied in the stated order because changing the order of operations may produce a different answer.

The original number is:

Its digits in ascending order are:

Compare corresponding positions:

• Position 1:
• Position 2:
• Position 3:
• Position 4:
• Position 5:
• Position 6:
• Position 7:
• Position 8:
• Position 9:

Only the digit remains in its original position. Therefore, the required number of unchanged digits is 1.

The original number is:

The required interchanges are:

• First and ninth digits:
• Second and eighth digits:
• Third and seventh digits:

The middle digits remain unchanged. The resulting number is:

Its first five-digit part is , and its last four-digit part is .

Therefore, the required difference is:

Hence, the resulting number is 182935467, and the required difference is 12826.

For two digits and , the number of digits between them in natural order is . In the given number, it is the difference between their positions minus .

Write the positions:

• , , , ,
• , , ,

A pair qualifies when:

The qualifying pairs are:

• : position difference , digit difference does not qualify, so it is excluded.
• : position difference , digit difference does not qualify.
• : position difference , digit difference , so it qualifies.
• : position difference , digit difference does not qualify.
• : position difference , digit difference , so it qualifies.
• : position difference , digit difference does not qualify.
• : position difference , digit difference , so it qualifies.

Thus, the qualifying pairs are , , and . Therefore, the total number of pairs is 3.

A ranking test determines the position of a person or object in an ordered row according to height, marks, age, performance, or another criterion.

• Rank from the top: The number of positions counted from the first or highest position to the person, including that person.
• Rank from the bottom: The number of positions counted from the last or lowest position to the person, including that person.
• Total number of persons: If the same person's ranks from both ends are known, the total is:

Here, is the rank from the top and is the rank from the bottom. One is subtracted because the same person is counted in both ranks.

For example, if a student is th from the top and th from the bottom, then:

Let the rank from the top be and the rank from the bottom be .

The total number of students is calculated by:

Substituting the values:

One is subtracted because the selected student is included once while counting from the top and again while counting from the bottom. Without subtraction, that student would be counted twice.

Therefore, the class contains 44 students.

A is th from the left.

Convert B's rank from the right into a rank from the left:

Thus, A is at position and B is at position from the left.

The number of students between them is:

The subtraction of excludes A and B themselves. Therefore, 15 students are positioned between A and B.

First convert Mohan's rank from the bottom to his rank from the top:

Mohan is therefore th from the top.

Ravi is ranks ahead of Mohan, so Ravi's rank from the top is:

Now convert Ravi's rank from the top to his rank from the bottom:

Therefore, Ravi is 15th from the top and 26th from the bottom.

A time sequence test requires events, activities, dates, days, months, or stages to be arranged in chronological order according to given conditions.

The principal steps are:

• Identify all events and time references.
• Translate words such as before, after, immediately before, two days later, and between into ordering constraints.
• Place fixed events first.
• Create a timeline or a table of available positions.
• Apply direct conditions before indirect conditions.
• Combine linked conditions into blocks where appropriate.
• Check that no event occupies more than one position.
• Verify the final order against every original condition.

For example, if A occurs before B and B occurs before C, then by transitivity the sequence is .

The five days are Monday, Tuesday, Wednesday, Thursday, and Friday.

Given:

• Q occurs immediately after S, forming the block .
• T occurs on Wednesday.
• R occurs on Friday.

The available days for P, S, and Q are Monday, Tuesday, and Thursday. Since S and Q must be consecutive, they can only occupy Monday and Tuesday. Therefore:

• Monday: S
• Tuesday: Q
• Wednesday: T
• Thursday: P
• Friday: R

P occurs before R, so all conditions are satisfied.

Hence, the only possible schedule is:

Let the stages be Planning , Design , Coding , Testing , and Deployment .

The conditions give:

• Design occurs after Planning: .
• Design occurs before Coding: .
• Testing occurs immediately after Coding: the block is .

Combining these conditions gives:

Deployment is the only remaining stage. Assuming deployment takes place after testing, as required by the normal project sequence, the complete arrangement is:

Thus, the order is:

1. Planning
2. Design
3. Coding
4. Testing
5. Deployment

The explicit conditions determine the first four positions, while the functional meaning of deployment places it after successful testing.

The days are Monday through Saturday.

Given:

• A is on Monday.
• E is on Tuesday.
• .
• B presents immediately after D, forming the block .

After placing A and E, the available days are Wednesday, Thursday, Friday, and Saturday for B, C, D, and F.

The block can occupy either Wednesday-Thursday, Thursday-Friday, or Friday-Saturday.

• If is Wednesday-Thursday, C and F occupy Friday and Saturday respectively. This gives .
• If is Thursday-Friday, C must be Wednesday and F must be Saturday. This gives .
• If is Friday-Saturday, C can be Wednesday or Thursday, but F must occur after C and no free day remains after the block. Therefore, this placement is impossible.

Hence, two schedules are possible:

• Monday to Saturday: A, E, D, B, C, F
• Monday to Saturday: A, E, C, D, B, F

The information does not determine one unique schedule.