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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.

ຕົວຢ່າງ​ທີ່​ໄດ້​ເຮັດ · 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)

ຂັ້ນ​ຕອນ​ຕໍ່​ຂັ້ນ​ຕອນ

  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).

ເປີດ​ເຜີຍ​ຄຳ​ຕອບ
\left(p \Rightarrow q\right) \Leftrightarrow \left(q \vee \neg p\right)

ປັດຈຸບັນ​ທ່ານ ເລືອກ​ບັນຫາ ຫຼື ພິມ ຫຼື ແຕ້ມ​ເອງ. ທຸກ​ບາດກ້າວ, ຮູບ, ຄໍາຕອບ​ຖືກ​ລົບ​ຈົນ​ກວ່າ​ເຈົ້າ​ຈະ​ຖາມ​ມັນ.

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ວິທີການ: 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 ∎.

ຄໍາຖາມທີ່ຄົນຖາມ

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.

ເພີ່ມເຕີມໃນ Set Theory & Logic

Sets and operationsRelations, functions and equivalenceCardinality and infinity