Step 2: The Grammar of Logic: Propositions, Truth Values & Implications
Before proving theorems, we must understand the atomic building blocks of mathematical arguments.
1. Definition of a Proposition (Mathematical Statement)
A Proposition is a declarative sentence that is either strictly True or strictly False, but cannot be both, and cannot be open to subjective interpretation.
- Valid Propositions:
- "New Delhi is the capital of India." $\implies$ True.
- "The square of any real number is non-negative ($x^2 \ge 0$)." $\implies$ True.
- "Every odd integer is divisible by 3." $\implies$ False (Counterexample: 5, 7, 11).
- Non-Propositions:
- "Will it rain tomorrow?" (Interrogative question).
- "Study mathematics diligently!" (Imperative command).
- "Geometry is more fascinating than algebra." (Subjective value judgment).
- "$x - 4 = 10$" (Open sentence; truth value unknown until $x$ is bound).
2. Conditional Propositions (Implications)
A statement of the form:
$$\mathbf{P \implies Q} \quad \text{("If } P \text{, then } Q\text{")}$$
- $P$ (Antecedent / Hypothesis): The premise or given condition.
- $Q$ (Consequent / Conclusion): The asserted consequence.
3. Truth Table of Conditional Statement ($P \implies Q$)
| $P$ (Hypothesis) | $Q$ (Conclusion) | $P \implies Q$ (Implication) | Real-World Analogy |
|---|---|---|---|
| T | T | T | Promise kept: You study hard (T), you score high (T). |
| T | F | F | Promise broken: You study hard (T), but score low (F). |
| F | T | T | Vacuously true: You didn't study (F), but still scored high (T). |
| F | F | T | Vacuously true: You didn't study (F), and didn't score high (F). |
[!NOTE] An implication $P \implies Q$ is False only when a True hypothesis leads to a False conclusion!