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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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · 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)

Àwọn ìṣàmúlò-ètò

\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).

Fi àwọn àgbèwọlé hàn

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

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Bii o ṣe le: 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 ∎.

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Set Theory & Logic

Sets and operations Relations, functions and equivalence Cardinality and infinity