Unit 2: Syllogism and Number Ranking - Subjective Questions

A syllogism is a form of deductive reasoning in which a conclusion is examined on the basis of two or more given statements called premises.

Basic components:

• Major premise: A general statement establishing a relationship between two classes.
• Minor premise: A statement introducing another class or specifying part of the major premise.
• Conclusion: A proposition claimed to follow logically from the premises.
• Terms: The subject and predicate classes occurring in the statements.

Example:

• All teachers are educated.
• Some women are teachers.
• Therefore, some women are educated.

The conclusion is valid because the women who are teachers must belong to the class of educated people. A syllogistic conclusion must follow from the premises alone; outside knowledge must not be used.

The four standard categorical propositions are:

• Universal affirmative — All A are B: The circle representing A lies completely inside B. It means .
• Universal negative — No A is B: The circles A and B do not overlap. It means .
• Particular affirmative — Some A are B: The circles A and B overlap, and at least one element exists in the common region. It means .
• Particular negative — Some A are not B: At least one element lies in the part of A outside B. It means .

Venn diagrams translate verbal relationships into spatial relationships, making it easier to determine whether a conclusion necessarily follows from the statements.

The Venn-diagram method can be applied as follows:

1. Identify the classes mentioned in the statements.
2. Draw a circle for each class.
3. Represent universal statements first:
   • Place one circle inside another for All A are B.
   • Keep two circles separate for No A is B.
4. Represent particular statements next:
   • Mark an element in an overlapping region for Some A are B.
   • Mark an element in the non-overlapping part of A for Some A are not B.
5. Examine whether the proposed conclusion is true in every valid arrangement of the diagram.

A conclusion follows only if it is forced by the statements. If it is true in one arrangement but false in another, it is only a possibility and not a definite conclusion.

The natural hierarchical relationship is:

• All roses are flowers.
• All flowers are plants.

Therefore, the circle for roses must lie entirely inside the circle for flowers, and the flower circle must lie entirely inside the circle for plants.

Symbolically:

From this arrangement, the following conclusions follow:

• All roses are flowers.
• All flowers are plants.
• All roses are plants.

The reverse relationships do not necessarily follow. For example, all plants need not be flowers, and all flowers need not be roses.

The statement All A are B fixes A completely inside B. However, Some B are C only requires B and C to overlap; it does not specify where that overlap occurs relative to A.

Consequently, several diagrams are possible:

• C may overlap both A and B, making Some A are C true.
• C may overlap the portion of B outside A, making No A is C true.
• C may contain A while also overlapping the remaining part of B.

Thus, neither Some A are C nor No A is C is a definite conclusion. Each may be possible because the premises leave the relationship between A and C undetermined.

This demonstrates that a valid diagram must satisfy all given statements, but unspecified relationships may be represented in multiple ways.

A possibility-based syllogism question asks whether a proposed relationship can exist without contradicting any given statement. Unlike a definite conclusion, the relationship need not hold in every valid diagram.

Method:

• Draw or imagine all relationships forced by the premises.
• Add the proposed possibility to the diagram.
• Check whether the resulting arrangement violates any premise.
• If at least one consistent diagram can be formed, the conclusion is possible.
• If every attempted arrangement creates a contradiction, it is impossible.

For example, from All A are B and No B is C, the possibility Some A are C is impossible because every A belongs to B, while B and C are disjoint. However, Some B are not A may be possible because the premises do not state that every B is A.

Let poets be P, dreamers be D, musicians be M, and athletes be A.

The statements give:

Conclusion 1: Some poets are musicians.

• This does not follow. The dreamers who are musicians need not be poets.

Conclusion 2: No poet is an athlete.

• This does not follow. Only musicians are known to be separate from athletes. Poets who are not musicians may be athletes.

Conclusion 3: Some dreamers are not athletes.

• This follows. Some dreamers are musicians, and no musician is an athlete. Therefore, those dreamers who are musicians are not athletes.

Result: Only conclusion 3 follows.

Let E represent engineers, G graduates, A artists, and P pilots.

The fixed relationships are:

Possibility 1: Some engineers are artists.

• Possible. The engineering circle may overlap the artist portion of graduates without violating any statement.

Possibility 2: No engineer is an artist.

• Possible. Engineers may lie entirely in the non-artist portion of the graduate class.

Possibility 3: Some engineers are pilots.

• Possible. Engineers who are not artists may be pilots because only artists are prohibited from being pilots.

Result: All three relationships are possible, although none is a definite conclusion from the premises.

In standard reasoning questions, Only a few books are journals conveys two facts:

• Some books are journals.
• Some books are not journals.

Given that All journals are indexed, the journal class lies completely within the indexed class.

Therefore:

• Some books are indexed definitely follows because some books are journals and every journal is indexed.
• Some books are not journals definitely follows directly from the phrase only a few.
• Some books are not indexed does not definitely follow. Books that are not journals could still be indexed for another reason.
• All books are indexed also does not follow because the indexing status of non-journal books is unknown.

Thus, only the first two conclusions are definite.

A definite conclusion must be true in every diagram satisfying the premises. A possible conclusion needs to be true in at least one valid diagram and must not contradict the premises.

Definite conclusion example:

• All doctors are educated.
• Some surgeons are doctors.
• Therefore, some surgeons are educated.

The conclusion is forced because the surgeons who are doctors must be educated.

Possible conclusion example:

• All doctors are educated.
• Some teachers are educated.
• Therefore, some doctors being teachers is possible.

The premises do not establish an overlap between doctors and teachers, but they also do not prohibit it. Hence, the overlap is possible but not definite.

The key distinction is necessity versus consistency.

A number ranking test evaluates the ability to determine the position of a person or object in an ordered arrangement.

Common information includes:

• Rank from the top or left.
• Rank from the bottom or right.
• Total number of persons or objects.
• Number of persons between two positions.
• Relative positions, such as one person being a fixed number of places above or below another.
• Changes caused by interchanging positions.

For a sequence containing persons, if a person is ranked from the left and from the right, then:

The subtraction of 1 prevents the same person from being counted twice.

Suppose a person is ranked from the left and from the right.

• The number of persons to the person's left is .
• The number of persons to the person's right is .
• Including the person, the total is:
  

Therefore:

Example: A student is 17th from the top and 24th from the bottom.

Using the formula:

Hence, there are 40 students in the class. The subtraction of 1 is necessary because the selected student is included in both ranks.

The relation among total students , rank from the top , and rank from the bottom is:

Rearranging for the rank from the bottom:

Substituting and :

Therefore, Riya is 36th from the bottom.

There are students above her and students below her. Including Riya gives students.

Initially:

• Aman is 16th from the top.
• Bharat is 21st from the top.

After interchanging positions, Aman occupies Bharat's original position. Therefore, Aman's new rank is 21st from the top. It is also given as 27th from the bottom.

Hence, the total number of students is:

Bharat now occupies Aman's original position, so Bharat is 16th from the top. His rank from the bottom is:

Answer:

• Total students: 47
• Bharat's new rank from the bottom: 32nd

If two persons occupy positions and from the same end, where , the number of persons strictly between them is:

For the 11th and 29th positions:

Therefore, 17 persons are between the two positions.

The subtraction of 1 excludes both endpoint positions. The persons between them occupy positions 12 through 28, which gives 17 positions in total.

First convert Q's rank from the right into a rank from the left:

Thus, P is at position 12 and Q is at position 32 from the left.

The number of students between them is:

Their midpoint position is:

Therefore, a student exactly midway between them is 22nd from the left. That student's rank from the right is:

Answer:

• Q is 32nd from the left.
• 19 students are between P and Q.
• The midway student is 22nd from the left and 19th from the right.

Write the adjacent digit pairs of :

(a) Even digits immediately preceded by an odd digit:

There are 4 such even digits.

(b) Odd digits immediately followed by an even digit:
The same four transitions satisfy this condition: , , , and .

There are therefore 4 such odd digits.

This type of number test requires checking adjacent digits while preserving their original order.

For the number to be greater than , its hundreds digit must be either or . For it to be even, its units digit must be either or .

Case 1: Hundreds digit is 3

• Units digit can be 2 or 4: 2 choices.
• After selecting the hundreds and units digits, the tens digit has 2 choices.
• Number of possibilities: .

They are and .

Case 2: Hundreds digit is 4

• The units digit must be 2 because repetition is not allowed.
• The tens digit can be 1 or 3: 2 choices.

They are and .

Thus, the total number is:

Hence, 6 three-digit even numbers greater than 300 can be formed.

Number test problems focus on the properties, order, or arrangement of digits and numbers. They may involve:

• Odd and even digits.
• Divisibility.
• Adjacent digit relationships.
• Rearrangement of digits.
• Formation of numbers under stated conditions.

The main strategy is to translate each verbal condition into a precise numerical restriction and examine digits systematically.

Ranking test problems focus on positions in a sequence. They may involve:

• Ranks from opposite ends.
• Relative positions.
• Interchange of positions.
• Persons between two ranks.

Useful ranking formulas include:

Number tests primarily require numerical pattern analysis, whereas ranking tests require careful positional counting.

A is 18th from the top. Since B is 7 places below A:

C is 12th from the bottom, so C's rank from the top is:

The number of candidates between B and C is:

After B and C interchange positions:

• B moves to C's original position, which is 39th from the top.
• C moves to B's original position, which is 25th from the top.

B's new rank from the bottom is:

C's new rank from the bottom is:

Answer:

• B was originally 25th from the top.
• C was originally 39th from the top.
• 13 candidates were between them.
• After interchange, B is 12th from the bottom and C is 26th from the bottom.