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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.

Eżempju maħdum: 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)

Pass b'pass

  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).

Jiżvelaw it-tweġiba
\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.

Ipprova tiegħek stess

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