Sets, Relations, Arguments

Question 1

Binary Relation

Answer 1

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

Question 2

Types of binary relation

Answer 2

A binary relation R is

(i) reflexive on a set S iff for all elements d of S the pair ⟨d, d⟩ is an element of R;
(ii) symmetric on a set S iff for all elements d, e of S: if ⟨d, e⟩ ∈ R then ⟨e, d⟩ ∈ R;
(iii) asymmetric on a set S iff for no elements d, e of S: ⟨d, e⟩ ∈ R and ⟨e, d⟩ ∈ R;
(iv) antisymmetric on a set S iff for no two distinct elements d, e of S: ⟨d, e⟩ ∈ R and ⟨e, d⟩ ∈ R;
(v) transitive on a set S iff for all elements d, e, f of S: if ⟨d, e⟩ ∈ R and ⟨e, f⟩ ∈ R, then ⟨d, f⟩∈R.

Question 3

Binary relations simpliciter

Answer 3

A binary relation R is

(i) symmetric iff it is symmetric on all sets;
(ii) assymmetric iff it is asymmetric on all sets;
(iii) antisymmetric iff it is antisymmetric on all sets;
(iv) transitive iff it is transitive on all sets.

Question 4

Equivalence relation

Answer 4

A binary relation R is an equivalence relation on S iff R is reflexive on S, symmetric on S and transitive on S.

Question 5

Function

Answer 5

A binary relation R is a function iff for all d, e, f: if ⟨d, e⟩ ∈ R and ⟨d, f⟩ ∈ R then e = f.

Question 6

Domain, range, into

Answer 6

(i) The domain of a function R is the set {d : there is an e such that ⟨d, e⟩ ∈ R}.
(ii) The range of a function R is the set {e : there is a d such that ⟨d, e⟩ ∈ R}.
(iii) R is a function into the set M iff all elements of the range of the function are in M.

Question 7

Function notation

Answer 7

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

Question 8

n-ary relation

Answer 8

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

Question 9

Argument

Answer 9

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

Question 10

Logical validity

Answer 10

An argument is logically valid iff there is no interpretation under which the premises are all true and the conclusion false.

Question 11

Logical consistency

Answer 11

A set of sentences is logically consistent iff there is at least one interpretation under which all sentences of the set are true.

Question 12

Logical truth

Answer 12

A sentence is logically true iff it is true under any interpretation.

Question 13

Contradiction

Answer 13

A sentence is a contradiction iff it is false under all interpretations.

Question 14

Logical equivalence

Answer 14

Sentences are logically equivalent iff they are true under exactly the same interpretations.