Chapter 1: Sets, Relations, and Arguments

Binary relation

A set is a binary relation iff it contains only ordered pairs.

A binary relation R is reflective on a set S…

Iff for all elements d or S the pair ⟨d, d⟩ is an element of R.

A binary relation R is symmetric on a set S… It is symmetric iff…

iff for all elements d, e of S: if ⟨d, e⟩ ∈ R then ⟨e, d⟩ ∈ R.

It fulfills the above on all sets.

A binary relation R is asymmetric on a set S… It is asymmetric iff…

iff for no elements d, e of S: ⟨d, e⟩ ∈ R and ⟨e, d⟩ ∈ R.

It fulfills the above on all sets.

A binary relation R is antisymmetric on a set S… It is antisymmetric iff…

iff for no two distinct elements d, e of S: ⟨d, e⟩ ∈ R and ⟨e, d⟩ ∈ R.

It fulfills the above on all sets.

A binary relation R is transitive on a set S… It is transitive iff…

iff for all elements d, e, f of S: if ⟨d, e⟩ ∈ R and ⟨e, f⟩ ∈ R, then ⟨d, f⟩ ∈ R.

It fulfills the above on all sets.

A binary relation R is an equivalence relation on S iff…

R is reflexive on S, symmetric on S, and transitive on S.

A binary relation R is a function…

iff for all d, e, f: if ⟨d, e⟩ ∈ R and ⟨d, f⟩ ∈ R then e = f.

Domain of a function

The domain of a function R is the set {d : there is an e such that ⟨d, e⟩ ∈ R}.

Range of a function

The range of a function R is the set {e : there is a d such that ⟨d, e⟩ ∈ R}.

R is a function into the set M iff…

All elements of the range of the function are in M.

Function notation

If d is in the domain of a function R one writes R(d) for the unique object e such that ⟨d, e⟩ is in R.

n-ary relation

An n-place relation is a set containing only n-tuples. An n-place relation is called a relation of arity n.

An argument…

Consists of a set of declarative sentences (the premises) and a declarative sentence (the conclusion) marked as the concluded sentence.

An argument is logically valid iff…

There is no interpretation under which the premises are all true and the conclusion false.

A set of sentences is logically consistent iff…

There is at least one interpretation under which all sentences of the set are true.

A sentence is logically true iff…

It is true under any interpretation.

A sentence is a contradiction…

iff it is false under all interpretations.

Sentences are logically equivalent iff…

They are true under exactly the same interpretations.