Sets, Relations, Arguments Flashcards
When is a set a binary relation?
A set is a binary relation if and only if it contains only ordered pairs.
When is a binary relation reflexive on a set S?
A binary relation R is reflexive on a set S iff for all d in S, the pair <d, d> is an element of R.
When is a binary relation symmetric?
A binary relation is symmetric iff for all d, e: if <d, e> ∈ R then <e, d> ∈ R.
When is a binary relation asymmetric?
A binary relation R is asymmetric iff for no <d, e>: <d, e> ∈ R and <e, d> ∈ R.
When is a binary relation antisymmetric?
A binary relation R is antisymmetric iff for no two distinct d, e: <d, e> ∈ R and <e, d› ∈ R.
When is a binary relation transitive?
A binary relation R is transitive iff for all d, e, f: if <d, e> ∈ R and <e, f> ∈ R, then also <d, f> ∈ R.
When is a binary relation an equivalence relation on a set S?
A binary relation R is an equivalence relation on S iff R is reflexive on S, symmetric and transitive.
When is a binary relation a function?
A binary relation R is a function iff for all d, e, f: if <d, e> ∈ R and <d, f> ∈ R then e=f.
What is the definition of i) the domain of a function ii) the range of a function iii) when R is a function into the set M?
(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 if and only if all elements of the range of the function are in M
How should a function be expressed?
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.
What is an argument?
An argument consists of a set of declarative sentences (the premises) and a declarative sentence (the conclusion) marked as the concluded sentence.
What is a characterisation of logical validity?
An argument is logically valid if and only if there is no interpretation under which the premises are all true and the conclusion is false.
What is a characterisation of consistency?
A set of sentences is consistent if and only if there is a least one interpretation under which all sentences of the set are true.
What is a characterisation of logical truth?
A sentence is logically true if and only if it is true under any interpretation.
What is a characterisation of a contradiction?
A sentence is a contradiction if and only if it is false under any interpretation.
What is a characterisation of logical equivalence?
Sentences are logically equivalent if and only if they are true under exactly the same interpretations.
