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Logic and methods of proof
Connectives, quantifiers, direct proof, contrapositive, contradiction, induction.
A proof is a chain of statements each following from earlier ones by a rule of inference. Direct proof assumes the hypothesis and derives the conclusion; contrapositive proves ¬Q → ¬P instead; contradiction assumes ¬Q and finds an absurdity; induction climbs the integers. Picture it: the truth table below, p → q and ¬p ∨ q agree in every row, which is the contrapositive method justified. Think it: Gödel showed any consistent system rich enough for arithmetic has true statements it cannot prove.
ตัวอย่างที่ทำงานแล้ว · truth table of (p implies q) iff (not p or q)
Truth table of (p implies q) iff (not p or q)
ก้าวต่อก้าว
- \left(p \Rightarrow q\right) \Leftrightarrow \left(q \vee \neg p\right)
2 variable(s) → 4 rows. Fill in every combination.
- \
A tautology (always true).
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วิธีการ: Logic and methods of proof
- Write the statement as "if P then Q" with P and Q precise.
- Choose a method: direct (assume P), contrapositive (assume not Q), contradiction (assume P and not Q), or induction (for all n).
- Each line must cite a definition, an axiom or a previous line.
- End when Q appears; mark it ∎.
คำถามที่คนถาม
Are some infinities bigger than others?
Yes. The integers and the rationals can be listed; the real numbers cannot (Cantor's diagonal argument), so there are strictly more reals than integers.
What is the difference between a relation and a function?
A relation pairs inputs with outputs freely; a function is a relation in which every input gets exactly one output.
เพิ่มเติมใน Set Theory & Logic
Sets and operationsRelations, functions and equivalenceCardinality and infinity