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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.
דוגמה עובדת · truth table of (p implies q) iff (not p or q)
Truth table of (p implies q) iff (not p or q)
צעד אחר צעד
- \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).
גלה את התשובה
עכשיו אתה בחר בעיה, או הקלד או צייר בעצמך. כל צעד, תמונה, התשובה חבויה עד שתשאל.
סמלים המשמשים כאן
הקש כל סמל להגדרה המלאה, תמונה, ומה כל אות בזה אומר.
איך: 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 ∎.
שאלות שאנשים שואלים
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.
יותר בפנים. Set Theory & Logic
Sets and operationsRelations, functions and equivalenceCardinality and infinity