Advanced Logical & Analytical Reasoning — Worked Practice

3A.1 Syllogism: A Worked Example with Two Premises

Premises: "All managers are graduates. Some graduates are engineers." From these two premises alone, no conclusion can validly connect managers and engineers, because the "some graduates" who are engineers are not guaranteed to overlap with the "all managers" who are graduates — the middle term "graduates" is not distributed in a way that forces the two end terms together. The only valid conclusion drawable is one that restates a given premise or a direct, immediate inference from it (for example, "some graduates are managers" is not directly derivable either, since "all managers are graduates" only guarantees the reverse is not universally false, not that it is true).

3A.2 Syllogism: A Worked Example with a Valid Conclusion

Premises: "All auditors are accountants. All accountants are professionals." Here the middle term "accountants" is fully distributed across both premises (via the universal "all" in each), which permits the valid chained conclusion: "All auditors are professionals." This is the standard universal-syllogism pattern (All A are B; All B are C; therefore All A are C) and is the most frequently tested valid form in the exam.

3A.3 Inequality Chains: A Worked Example

Given: P > Q ≥ R > S and R ≥ T. To determine whether "P > T" necessarily follows, chain the inequalities: P > Q, and Q ≥ R, so P > R (since a strict inequality combined with a "greater than or equal to" preserves the strict inequality). Since R ≥ T, and P > R, it follows that P > T. This illustrates the key rule: a chain of inequalities combining ">" and "≥" produces a strict "> " conclusion overall as long as at least one link in the chain is strict.

3A.4 Data Sufficiency: A Worked Example

Question: "What is the value of y?" Statement I: 2y − 6 = 14. Statement II: y is a positive even number. Statement I alone gives 2y = 20, so y = 10, which is a single definite value — Statement I alone is sufficient. Statement II alone only narrows y to positive even numbers (2, 4, 6, 8, 10, ...), which is not a single value — Statement II alone is not sufficient. Since Statement I alone answers the question, the correct classification is "Statement I alone is sufficient, Statement II alone is not."

3A.5 Coding-Decoding: A Worked Letter-Shift Example

If each letter of a word is shifted forward by two positions in the alphabet, then WATER becomes YCVGT (W→Y, A→C, T→V, E→G, R→T). Applying the same two-position forward shift to MOTOR gives OQVQT (M→O, O→Q, T→V, O→Q, R→T). The key discipline in coding-decoding is to apply the identified shift or rule to every letter independently and verify the transformation against the given example before applying it to the new word.

3A.6 Blood Relations: A Worked Multi-Generational Example

"A is the father of B. B is the sister of C. C is the husband of D. D is the mother of E." Working through generation by generation: A is B's father, so A is one generation above B. B and C are siblings, so C is also A's child, one generation below A. C is D's husband, placing D in C's generation (not A's descendant by blood, but married in). D is E's mother, so E is one generation below C and D. Since C is A's child and E is C's child, A is E's grandfather.

3A.7 Practice Set — Advanced Logical Reasoning (16 MCQs)

1. Premises: "All auditors are accountants. All accountants are professionals." Which conclusion follows?
   (a) All auditors are professionals (b) All professionals are auditors (c) No auditors are professionals (d) Some professionals are not auditors
2. Premises: "All managers are graduates. Some graduates are engineers." Which conclusion follows?
   (a) No definite conclusion connecting managers and engineers can be drawn (b) All managers are engineers (c) All engineers are managers (d) No managers are engineers
3. Given P > Q ≥ R > S and R ≥ T, does P > T necessarily follow?
   (a) Yes, P > T follows validly (b) No, it cannot be determined (c) Only if Q = R (d) Only if S = T
4. Data Sufficiency: "What is the value of y?" Statement I: 2y − 6 = 14. Statement II: y is a positive even number. Which is sufficient?
   (a) Statement I alone is sufficient, Statement II alone is not (b) Statement II alone is sufficient, Statement I alone is not (c) Both together are necessary (d) Neither is sufficient even together
5. If each letter is shifted forward by two positions, WATER becomes YCVGT. Using the same rule, MOTOR becomes:
   (a) OQVQT (b) OQVPT (c) NQVQT (d) OPVQT
6. A is the father of B. B is the sister of C. C is the husband of D. D is the mother of E. How is A related to E?
   (a) Grandfather (b) Father (c) Uncle (d) Great-grandfather
7. Premises: "No trainers are students. Some students are athletes." Which conclusion follows?
   (a) Some athletes are not trainers (b) All athletes are trainers (c) No athletes are trainers (d) All trainers are athletes
8. Given: X ≥ Y > Z and Z ≥ W. Which relationship between X and W necessarily holds?
   (a) X > W (b) X < W (c) X = W (d) Cannot be determined
9. Data Sufficiency: "Is n an odd number?" Statement I: n is divisible by 3. Statement II: n + 1 is even. Which is sufficient?
   (a) Statement II alone is sufficient, Statement I alone is not (b) Statement I alone is sufficient, Statement II alone is not (c) Both together are required (d) Neither is sufficient even together
10. If each letter is shifted backward by three positions, FROG becomes COLD. Using the same rule, PLANT becomes:
    (a) MIXKQ (b) MIXKR (c) MIYKQ (d) NIXKQ
11. P is the mother of Q. Q is the brother of R. R is the wife of S. S is the father of T. How is P related to T?
    (a) Grandmother (b) Mother (c) Aunt (d) Sister
12. Premises: "All coins are metals. Some metals are magnetic." Which conclusion follows?
    (a) No definite conclusion connecting coins and magnetism can be drawn (b) All coins are magnetic (c) No coins are magnetic (d) All magnetic things are coins
13. Given: M > N ≥ O and O > P ≥ Q. Which relationship between M and Q necessarily holds?
    (a) M > Q (b) M < Q (c) M = Q (d) Cannot be determined
14. Data Sufficiency: "What is the value of z?" Statement I: z² = 49. Statement II: z is a positive number. Which is sufficient?
    (a) Both statements together are sufficient, but neither alone is (b) Statement I alone is sufficient (c) Statement II alone is sufficient (d) Neither is sufficient even together
15. If each letter is shifted forward by one position, CAT becomes DBU. Using the same rule, HORSE becomes:
    (a) IPSTF (b) IPSTE (c) IPSTG (d) IPRSF
16. A is the son of B. B is the daughter of C. D is the husband of B. How is D related to A?
    (a) Father (b) Uncle (c) Grandfather (d) Brother-in-law

Answer Key

1.(a)

2.(a)

3.(a)

4.(a)

5.(a)

6.(a)

7.(a)

8.(a)

9.(a)

10.(a)

11.(a)

12.(a)

13.(a)

14.(a)

15.(a) 16.(a)