MATH 144 FINAL EXAM 78 QUESTIONS WITH
VERIFIED ANSWERS
Equivalence on A - CORRECT ANSWER R is called an equivalence on An if it is
reflexive, symmetric & transitive.
Reflexive - CORRECT ANSWER If for all a ∈ A, area
Symmetric - CORRECT ANSWER If for all a, b ∈ A, garb implies bra
Transitive - CORRECT ANSWER If for all a, b, c ∈ A, garb & bark imply arc
Equivalence Class of a Modulo E - CORRECT ANSWER Let Me be an equivalence on
A & let a ∈ A. The equivalence class of a modulo e is the set
[a]↓E = { x ∈ A : exam }
Partition - CORRECT ANSWER A system S of nonempty sets is a mutually disjoint,
i.e.
a. C, D ∈ S & C ≠ D , then C ∩ D = ∅
b. union of S is the whole set A, i.e. U S = A
A/E - CORRECT ANSWER Let Me be an equivalence on A. The system of all
equivalence classes modulo E is
A/E = {[a]↓E : a ∈ a }
, E↓S - CORRECT ANSWER Let S be a partition of A.
E↓S = {(am) ∈ A x A , ∃C ∈ S : a ∈ C & b ∈ C
Set of Representatives - CORRECT ANSWER A set X ⊆ A is called .... for the
equivalence E↓S if for every C ∈ S, X ∩ C = {a} for some a ∈ C
Antisymmetric - CORRECT ANSWER Let R be a binary relation. If for all a, b ∈ A,
garb & bra imply a = b
(Partial) Ordering - CORRECT ANSWER Let R be a binary relation which is reflexive,
antisymmetric, & transitive.
Ordered Set - CORRECT ANSWER The pair (A,R)
≤ - CORRECT ANSWER Symbol used to denote ordering.
Asymmetric - CORRECT ANSWER Let S be a relation on A. If as implies that base
does not hold for any a, b ∈ A, i.e. as & base can never both be true.
Strict Ordering - CORRECT ANSWER If the relation Sin A is asymmetric & transitive.
Comparable - CORRECT ANSWER Let a, b ∈ A & let ≤ be an ordering of A. a & b are
comparable in the ordering ≤ if a ≤ b or b ≤ a
VERIFIED ANSWERS
Equivalence on A - CORRECT ANSWER R is called an equivalence on An if it is
reflexive, symmetric & transitive.
Reflexive - CORRECT ANSWER If for all a ∈ A, area
Symmetric - CORRECT ANSWER If for all a, b ∈ A, garb implies bra
Transitive - CORRECT ANSWER If for all a, b, c ∈ A, garb & bark imply arc
Equivalence Class of a Modulo E - CORRECT ANSWER Let Me be an equivalence on
A & let a ∈ A. The equivalence class of a modulo e is the set
[a]↓E = { x ∈ A : exam }
Partition - CORRECT ANSWER A system S of nonempty sets is a mutually disjoint,
i.e.
a. C, D ∈ S & C ≠ D , then C ∩ D = ∅
b. union of S is the whole set A, i.e. U S = A
A/E - CORRECT ANSWER Let Me be an equivalence on A. The system of all
equivalence classes modulo E is
A/E = {[a]↓E : a ∈ a }
, E↓S - CORRECT ANSWER Let S be a partition of A.
E↓S = {(am) ∈ A x A , ∃C ∈ S : a ∈ C & b ∈ C
Set of Representatives - CORRECT ANSWER A set X ⊆ A is called .... for the
equivalence E↓S if for every C ∈ S, X ∩ C = {a} for some a ∈ C
Antisymmetric - CORRECT ANSWER Let R be a binary relation. If for all a, b ∈ A,
garb & bra imply a = b
(Partial) Ordering - CORRECT ANSWER Let R be a binary relation which is reflexive,
antisymmetric, & transitive.
Ordered Set - CORRECT ANSWER The pair (A,R)
≤ - CORRECT ANSWER Symbol used to denote ordering.
Asymmetric - CORRECT ANSWER Let S be a relation on A. If as implies that base
does not hold for any a, b ∈ A, i.e. as & base can never both be true.
Strict Ordering - CORRECT ANSWER If the relation Sin A is asymmetric & transitive.
Comparable - CORRECT ANSWER Let a, b ∈ A & let ≤ be an ordering of A. a & b are
comparable in the ordering ≤ if a ≤ b or b ≤ a
