Logical Deductions: Sets, Quantifiers and Counterexamples

Scope and method

This lesson supports deduction within GATE General Aptitude analytical aptitude. It is original sectional practice, not an official question paper or complete GA mock. GATE is conducted in English; Hindi here supports learning. Official syllabus · Exam language

A conclusion follows only if it is true in every situation satisfying the premises. Treat the premises as given. Real-world plausibility cannot fill a missing logical link. One situation with true premises and a false conclusion disproves validity. OpenStax: Inferences

Translate before deciding

“All A are B” means A is contained in B; it does not reverse to “All B are A.” “No A are B” means their intersection is empty. “Some A are B” asserts at least one shared member; “some” does not mean exactly one.

Unless stated otherwise, a category may be empty. Universal statements alone do not establish that its members exist. The negation of “All A are B” is “Some A are not B.” It is not “No A are B.” Stanford: First-order logic

Worked example 1: chain and exclusion

Premises: all certified sensors are tested; no tested sensor is rejected. Every certified sensor lies in the tested group, which has no overlap with the rejected group. Therefore, no certified sensor is rejected. The conclusion remains valid if no certified sensors exist. “Some certified sensors are accepted” does not follow: existence and acceptance were never supplied.

Worked example 2: find a counterexample

Premises: all A are B; some B are C. Proposed conclusion: some A are C. Choose A = {1}, B = {1,2}, C = {2}. Both premises hold, yet A and C have no shared member. The proposed inference is invalid. These numbers are merely object labels. A diagram showing overlap would establish a possibility, not a necessity.

Worked example 3: use an existing witness

Premises: every red parcel is registered; some red parcels are fragile. Pick one red-and-fragile parcel guaranteed by the second premise. It must also be registered by the first. Therefore, some registered parcels are fragile. Nothing establishes that every registered parcel is fragile. Track the same witness through the argument.

A repeatable solving routine

Write the groups, translate each quantifier, mark known existing members, and test the proposed conclusion. Keep separate witnesses separate unless the premises identify them. Check the direction of every implication. For P implies Q, knowing Q alone does not establish P. Cornell: Logic

Self-checks

1. All A are B. Must any A exist? No. An empty A satisfies the premise.
2. Negate “Every badge is blue.” At least one badge is not blue. One exception defeats “every.”
3. If an alarm sounds, a lamp glows. The lamp glows. Must the alarm sound? No. The premise permits the lamp to glow for another reason.