1 Sets, Relations, and Arguments

Question 1

Define ‘Binary Relation’

Answer 1

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

Question 2

What makes binary relation R reflexive on a set S

Answer 2

R is reflexive on S iff for all elements d of S the pair <d,d> is an element of R

Question 3

What makes binary relation R symmetric on a set S

Answer 3

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

Question 4

What makes binary relation R asymmetric on a set S

Answer 4

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

Question 5

What makes binary relation R antisymmetric on a set S

Answer 5

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

Question 6

What makes binary relation R transitive on a set S

Answer 6

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

Question 7

Define an equivalence relation

Answer 7

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

Question 8

Give an example of an equivalence relation

Answer 8

Being born on the same day

Question 9

Define a function (for binary relations)

Answer 9

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)

Question 10

Define the domain of a function R (R is a binary relation)

Answer 10

The domain of a function R is the set {d : there is an e such that <d,e> is in R}

Question 11

Define the range of a function R (R is a binary relation)

Answer 11

The range of a function R is the set {e : there is a d such that <d,e> is in R}

Question 12

What makes R a function INTO a set

Answer 12

R is a function into the set M iff all elements of the range of the function are in M

Question 13

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

Answer 13

R(d)

Question 14

Define an n-ary relation

Answer 14

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

Question 15

What is arity

Answer 15

An n-place relation is called a relation of arity n

Question 16

Define an argument

Answer 16

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

Question 17

Define logical validity

Answer 17

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

Question 18

Define logical validity in terms of inconsistency

Answer 18

An argument is logically valid iff the set obtained by added the negation of the conclusion to the premisses is inconsistent

Question 19

Define consistency

Answer 19

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

Question 20

Define logical truth

Answer 20

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

Question 21

Define a contradiction

Answer 21

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

Question 22

Define logical equivalence

Answer 22

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

Question 23

Define a declarative sentence

Answer 23

A sentence which is either true of false.

Question 24

Define the empty set

Answer 24

The set which contains no elements

Question 25

Is the empty set a binary relation?

Answer 25

Yes, it contains no non-binary relations.

Question 26

Is the empty set reflexive?

Answer 26

Yes

Question 27

Is the empty set transitive?

Answer 27

Yes

Question 28

Is the empty set symmetric?

Answer 28

Yes

Question 29

IS the empty set asymmetric?

Answer 29

Yes

Question 30

Is the empty set anti-symmetric?

Answer 30

Yes

Question 31

List the 5 properties of the empty set

Answer 31

• Reflexive
• Symmetric
  -Asymmetric
• Anti-symmetric
• Transitive