Study Notes – Statement & Conclusion
(Reasoning Ability – SBI PO, IIT‑JEE style)
Definition
Statements – A set of propositions that are assumed to be true (unless a contradiction is revealed).
Conclusion – A proposition that has to be judged (i) true, (ii) false, or (iii) cannot be determined from the statements.
1. Core Idea
- All given statements are true – the only source of truth.
- Logical consequence – A conclusion is true iff every valuation that satisfies all statements also satisfies the conclusion.
- Contradiction – A conclusion is false iff every valuation that satisfies all statements makes the conclusion false.
- Indeterminate – If some valuations satisfy the statements & conclusion, and others satisfy the statements but not the conclusion, the conclusion is indeterminate.
Why this matters?
In competitive exams you must decide without enumerating all possible models. Use inference rules, truth tables for a few variables, or a “quick check” method.
2. Logical Notation & Symbols
| Symbol | Meaning | Example |
|---|---|---|
¬ |
NOT | ¬P = “P is false” |
∧ |
AND | P ∧ Q = “P and Q are true” |
∨ |
OR | P ∨ Q = “P or Q (or both) are true” |
→ |
IF … THEN | P → Q = “If P then Q” |
↔ |
IF AND ONLY IF | P ↔ Q = “P and Q have the same truth value” |
⊤ |
TRUE | |
⊥ |
FALSE |
Tip – Convert every statement into a well‑formed formula (WFF). Then you can manipulate it using rules of inference.
3. Key Rules of Inference (with Boolean Derivation)
| Rule | Condition | Consequence | Boolean Derivation |
|---|---|---|---|
| Modus Ponens (MP) | P → Q, P |
Q |
P → Q: if |