Unit 2: Syllogism and Number Ranking Test

I. Orientation

Analytical reasoning determines what must be true, may be true, or cannot be true from stated conditions. In this unit, syllogism uses relationships among classes or groups, while number and ranking tests use ordered data, positions, and numerical transformations.

• Governing principle: Use only the information explicitly supplied; real-world knowledge must not alter the stated relationships.
• Deductive validity: A conclusion is valid only when it follows in every arrangement satisfying the premises.
• Possibility: A possibility conclusion is acceptable when at least one arrangement satisfies it without contradicting any premise.
• Class convention: Capital letters such as (A), (B), and (C) represent groups or sets; an individual may belong to more than one set.
• Order convention: In ranking problems, positions are counted inclusively from the specified end.
• Operation convention: In number tests, perform transformations in the exact order stated and compare only the requested results.
• Verification principle: A diagram, table, or positional formula should confirm the conclusion rather than replace careful reading.

II. Syllogism — Reasoning About Class Relationships

A syllogism consists of statements called premises and a conclusion whose validity must be tested. Categorical syllogisms commonly use the forms “all,” “no,” and “some,” which can be represented through sets.

All A are B       : A ⊆ B
No A is B         : A ∩ B = ∅
Some A are B      : A ∩ B ≠ ∅
Some A are not B  : A − B ≠ ∅

Here, (A) and (B) are sets, (\subseteq) means “is a subset of,” (\cap) means intersection, (\varnothing) is the empty set, and (A-B) means members of (A) outside (B).

A. Logical Venn diagrams

Logical Venn diagrams translate verbal class relationships into spatial relationships so that conclusions can be checked visually.

• Universal affirmative: “All (A) are (B)” places the entire (A)-circle inside (B).
  

  TEXT

    B: (       A: (   )       )

  • It does not establish that all (B) are (A).
  • It does not, by itself, prove that any (A) exists.

• Universal negative: “No (A) is (B)” requires two non-overlapping regions.
  

  TEXT

    ( A )       ( B )

  • The relation is reversible: if no (A) is (B), then no (B) is (A).

• Particular affirmative: “Some (A) are (B)” places an existence mark in the common region.
  

  TEXT

    ( A  x  B )

  • The relation is reversible: some (B) are (A).
  • “Some” means at least one, not necessarily only one.

• Particular negative: “Some (A) are not (B)” places an existence mark in the part of (A) outside (B).

  • Its converse, “Some (B) are not (A),” does not necessarily follow.

• Chain relationship: From (A\subseteq B) and (B\subseteq C), it follows that (A\subseteq C).
  

  TEXT

    A ⊆ B, B ⊆ C  ⇒  A ⊆ C

• Disjointness through inclusion: If (A\subseteq B) and (B\cap C=\varnothing), then (A\cap C=\varnothing).

• Worked example: Consider “All poets are writers,” “No writer is illiterate,” and “Some poets exist.”

  • Let (P), (W), and (I) represent poets, writers, and illiterate persons.
  • The premises give (P\subseteq W), (W\cap I=\varnothing), and (P\neq\varnothing).
  • Therefore, (P\cap I=\varnothing): no poet is illiterate.
  • Since poets exist and (P\subseteq W), some writers are poets.

B. Ability to relate a certain given group diagrammatically

Relating a given group diagrammatically requires identifying whether the groups show inclusion, overlap, separation, or a combination of these relations.

• Complete inclusion: Use nested circles when every member of one class belongs to another.

  • Example: every square is a rectangle, so (\text{Squares}\subseteq\text{Rectangles}).

• Partial overlap: Use intersecting circles when some, but not necessarily all, members are common.

  • Example: some teachers are authors; neither class must contain the other.

• Complete separation: Use distinct circles when the groups cannot share members.

  • Example: even integers and odd integers are disjoint.

• One class within two overlapping classes: Place the smaller class inside the intersection when all its members belong to both larger classes.

  • If all surgeons are doctors and all surgeons are graduates, then surgeons lie in (\text{Doctors}\cap\text{Graduates}).
  • This does not establish a general relationship between all doctors and all graduates.

• Mixed inclusion and exclusion: If all (A) are (B), while no (B) is (C), draw (A) inside (B) and (C) separately.
  

  TEXT

    (( A ) B)        ( C )

• Ambiguous relationship: When no premise relates two classes, their exact relationship remains undetermined.

  • From “All (A) are (B)” and “All (C) are (B),” (A) and (C) may overlap, be separate, or one may contain the other.
  • A diagram must preserve these alternatives instead of assuming one arrangement.

• Diagram-selection method:

  1. Identify universal relations: “all” or “no.”
  2. Draw inclusion and exclusion boundaries.
  3. Add particular statements using existence marks.
  4. Check whether each proposed conclusion holds in every permitted diagram.

• Worked example: For animals, dogs, and cats, “All dogs are animals” and “All cats are animals” require two classes inside animals. Dogs and cats may be drawn separately only if an additional statement, such as “No dog is a cat,” establishes disjointness.

C. Possibility-based questions

Possibility-based questions ask whether a proposed relationship can exist in at least one diagram consistent with all the premises.

• Possibility standard: A conclusion is possible unless the premises directly or indirectly prohibit it.
  

  TEXT

    Possible(C) ⇔ Premises ∧ C is consistent

  
  Here, (C) is the proposed conclusion and (\land) means logical conjunction.

• Necessity versus possibility:

  1. A necessary conclusion must hold in every valid arrangement.
  2. A possible conclusion needs to hold in only one valid arrangement.

• Direct prohibition: “No (A) is (B)” makes “Some (A) are (B)” impossible because it requires (A\cap B) to be both empty and non-empty.

• Indirect prohibition: If all (A) are (B) and no (B) is (C), then some (A) being (C) is impossible.

• Unspecified overlap: If all (A) are (B) and all (C) are (B), some (A) being (C) is possible because no premise separates them.

• Reverse possibility: “All (A) are (B)” permits “Some (B) are not (A)” only when (B) is allowed to have members outside (A); it does not force such members to exist.

• Existence caution: Universal premises describe boundaries but do not necessarily assert that a class has members. Particular statements containing “some” explicitly assert existence.

• Worked example: Given “All dancers are artists” and “Some singers are artists,” the possibility that some singers are dancers is valid. The singer existence mark may be placed inside the dancer region, and neither premise forbids that overlap. However, the overlap is possible rather than necessary.

III. Number Tests — Ordered Numerical Operations

Number tests assess the ability to inspect numbers, apply specified digit-level operations, arrange results, and count values satisfying exact conditions. Accuracy depends on treating each number as an ordered sequence of digits.

A. Number test problems

Number test problems commonly involve digit comparison, rearrangement, replacement, divisibility, and counting after a transformation.

• Place value: In a number (abcd), the digits represent thousands, hundreds, tens, and units respectively.
  

  TEXT

    abcd = 1000a + 100b + 10c + d

• Digit reversal: A three-digit number (100a+10b+c) becomes (100c+10b+a).

  • Leading zeroes do not retain place value: reversing (240) gives (042=42).

• Digit interchange: When specified positions are exchanged, unchanged positions must remain fixed.

  • Swapping the first and third digits of (5724) gives (2754).

• Ascending and descending arrangement:

  • Ascending order places the smallest value first.
  • Descending order places the largest value first.
  • Repeated values retain separate positions when counting items.

• Divisibility checks:

  • A number is divisible by (2) if its final digit is even.
  • It is divisible by (3) if its digit sum is divisible by (3).
  • It is divisible by (5) if its final digit is (0) or (5).
  • It is divisible by (9) if its digit sum is divisible by (9).

• Conditional counting: Translate every condition separately before counting.

  • “First digit greater than the last” means (a>d).
  • “Exactly one even digit” excludes numbers containing zero or two or more even digits beyond the required count.

• Worked example: For (482, 731, 564,) and (293), reverse each number and arrange the results in ascending order.

  • Reversed values are (284, 137, 465,) and (392).
  • Ascending order is (137, 284, 392, 465).
  • The original number corresponding to the third transformed value is (293).

IV. Ranking Tests — Determining Relative Position

Ranking tests determine the position of a person or object from the top, bottom, left, or right, often by combining two relative descriptions.

A. Ranking test problems

Ranking test problems are solved by converting verbal positions into inclusive counts and applying the appropriate positional relationship.

• Total from two ends: If one person is (L)th from the left and (R)th from the right, the total number (N) is:
  

  TEXT

    N = L + R - 1

  
  The subtraction avoids counting the same person twice.

• Opposite-end rank: If there are (N) persons and a person is (L)th from the left:
  

  TEXT

    R = N - L + 1

  
  Here, (R) is the rank from the right.

• Persons between two ranks on the same side: For positions (p) and (q):
  

  TEXT

    B = |p - q| - 1

  
  Here, (B) is the number between them and (|p-q|) is the absolute difference.

• Cross-end conversion: Convert both positions to the same reference end before finding the gap.

  • In a row of (N), (r)th from the right equals ((N-r+1))th from the left.

• Interchange of positions: If two persons exchange positions, track each person’s old and new rank separately.

  • The movement distance is the absolute difference between the two positions.
  • The total size can be found when one person’s new rank is known from the opposite end.

• Tied ranks: In ordinary reasoning questions, ranks are usually unique unless equality is explicitly stated.

  • With competition ranking, two people tied at rank (2) may cause the next rank to be (4), so the stated convention controls the calculation.

• Worked example: Mira is (18)th from the left and (27)th from the right in a row.
  

  TEXT

    N = 18 + 27 - 1 = 44

  
  Therefore, the row contains (44) people. A person who is (12)th from the right would be (44-12+1=33)rd from the left.