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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.

tusaale · truth table of (p implies q) iff (not p or q)

Discrete-Math · Propositional-Logic

Truth table of (p implies q) iff (not p or q)

tallaabo tallaabo

  1. \left(p \Rightarrow q\right) \Leftrightarrow \left(q \vee \neg p\right)

    2 variable(s) → 4 rows. Fill in every combination.

  2. \

    A tautology (always true).

Jawaabta muuji
\left(p \Rightarrow q\right) \Leftrightarrow \left(q \vee \neg p\right)

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Ku hay shaqadaada

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Ku soo biir Ku soo gal

Calaamadaha la isticmaalay halkan

Taab calaamad kasta oo loogu talagalay qeexidda buuxda, sawir, iyo waxa xarfaha oo dhan ka mid ah ka dhigan tahay.

Sida loo: Logic and methods of proof

  1. Write the statement as "if P then Q" with P and Q precise.
  2. Choose a method: direct (assume P), contrapositive (assume not Q), contradiction (assume P and not Q), or induction (for all n).
  3. Each line must cite a definition, an axiom or a previous line.
  4. End when Q appears; mark it ∎.

Su'aalaha dadku waydiiyaan

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.

In ka badan Set Theory & Logic

Sets and operationsRelations, functions and equivalenceCardinality and infinity