1 Sets, Relations, and Arguments Flashcards
Define ‘Binary Relation’
A set is a binary relation iff it contains only ordered pairs
What makes binary relation R reflexive on a set S
R is reflexive on S iff for all elements d of S the pair <d,d> is an element of R
What makes binary relation R symmetric on a set S
R is symmetric on S iff for all elements d, e of S: if <d,e> is in R then <e,d> is in R
What makes binary relation R asymmetric on a set S
R is asymmetric on S iff for no elements d, e of S: if <d,e> is in R then <e,d> is in R
What makes binary relation R antisymmetric on a set S
R is antisymmetric on a set S iff for no two distinct elements d, e of S: if <d,e> is in R then <e, d> is in R
What makes binary relation R transitive on a set S
R is transitive on S iff for all elements d, e, f of S: if <d,e> and <e,f> are in R, then <d,f> is in R
Define an equivalence relation
A binary relation R is an equivalence relation on S iff R is reflexive on S, symmetric on S, and transitive on S
Give an example of an equivalence relation
Being born on the same day
Define a function (for binary relations)
A binary relation R is a function iff for all d, e, f: if <d,e> is in R and <d,f> is in R then e=f.
(for every d, there is at most one e)
Define the domain of a function R (R is a binary relation)
The domain of a function R is the set {d : there is an e such that <d,e> is in R}
Define the range of a function R (R is a binary relation)
The range of a function R is the set {e : there is a d such that <d,e> is in R}
What makes R a function INTO a set
R is a function into the set M iff all elements of the range of the function are in M
If d is in the domain of a function R, what would one write for the unique object e such that <d,e> is in R
R(d)
Define an n-ary relation
An n-place relation is a set containing only n-tuples.
What is arity
An n-place relation is called a relation of arity n
Define an argument
An argument consists of a set of declarative sentences (the premisses) and a declarative sentence (the conclusion) marked as the concluded sentence.
Define logical validity
An argument is logically valid iff there is no interpretation under which the premisses are all true and the conclusion false.
Define logical validity in terms of inconsistency
An argument is logically valid iff the set obtained by added the negation of the conclusion to the premisses is inconsistent
Define consistency
A set of sentences is logically consistent iff there is at least one interpretation under which all sentences of the set are true.
Define logical truth
A sentence is logically true iff it is true under any interpretation.
Define a contradiction
A sentence is a contradiction iff it is false under all interpretations.
Define logical equivalence
Sentences are logically equivalent iff they are true under exactly the same interpretations.
Define a declarative sentence
A sentence which is either true of false.
Define the empty set
The set which contains no elements
Is the empty set a binary relation?
Yes, it contains no non-binary relations.
Is the empty set reflexive?
Yes
Is the empty set transitive?
Yes
Is the empty set symmetric?
Yes
IS the empty set asymmetric?
Yes
Is the empty set anti-symmetric?
Yes
List the 5 properties of the empty set
- Reflexive
- Symmetric
-Asymmetric - Anti-symmetric
- Transitive
