maths.free › Set Theory & Logic › Logic and methods of proof

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.

Nhazi ahụ ejirila: 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)

Nzọụkwụ site n'ụdị

  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).

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

Symbols used here

\neg,\ \wedge,\ \vee,\ \Rightarrow,\ \Leftrightarrow
not, and, or, implies, iff
Logical connectives.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\neq
not equal
The two sides are different.
\infty
infinity
Not a number: "grows without bound" in limits and intervals.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
x \in A,\ A \subseteq B
element of, subset
x belongs to A; every element of A is in B.
A \cup B,\ A \cap B,\ A \setminus B
union, intersection, difference
In either; in both; in A but not B.
\emptyset,\ |A|
empty set, cardinality
The set with no elements; the number of elements of A.
\forall,\ \exists
for all, there exists
Quantifiers: every x; at least one x.
\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.

How to: 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 ∎.

Questions people ask

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.

Jiri gị onwe gị

Oge Set Theory & Logic

Sets and operationsRelations, functions and equivalenceCardinality and infinity