कक्षा 9–12Foundationsनियोजित
Propositional logic and truth tables
Truth tables, implication with its converse and contrapositive, logical equivalence and tautologies: the logic of 'and', 'or', 'not' and 'if…then' made exact.
math.found.propositional-logic
इस विषय का पाठ अभी लिखा नहीं गया है। नीचे दी गई रूपरेखा पाठ्यक्रम-मानचित्र का हिस्सा है और इसकी समीक्षा नहीं हुई है।
इस पर आधारित
4 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- Build truth tables for compound statements and use them to test logical equivalence.
- Distinguish an implication from its converse, inverse and contrapositive.
- Negate compound and quantified statements, including implications.
परिभाषा
- P → Q is false only when P is true and Q is false.
- Two formulas are logically equivalent if they have the same truth value under every assignment of truth values to their variables.
- A tautology is a formula that is true under every assignment.
प्रमेय
- P → Q is logically equivalent to its contrapositive ¬Q → ¬P, but not to its converse Q → P.
- De Morgan's laws: ¬(P ∧ Q) ≡ ¬P ∨ ¬Q and ¬(P ∨ Q) ≡ ¬P ∧ ¬Q.
- ¬(P → Q) ≡ P ∧ ¬Q.
आम भ्रांतियाँ
- Thinking an implication with a false hypothesis is false; it is true.
- Treating the converse of an implication as equivalent to it.
- Reading 'if P then Q' as claiming that P causes Q.
आगे ले जाता है
वैकल्पिक खोजें
- Computational complexity Deciding whether a propositional formula is satisfiable (SAT) was the first problem proved NP-complete (Cook–Levin theorem).
- Type theory and propositions as types In constructive logic and type theory, a proof of P → Q is a method turning proofs of P into proofs of Q, and P ∨ ¬P is not assumed as a law.
अधिगम पथ
आगे पढ़ने के लिए
- Daniel J. Velleman, How to Prove It: A Structured Approach (1994)
- Herbert B. Enderton, A Mathematical Introduction to Logic (1972)