Discrete Structures Final UPDATED
ACTUAL Exam Questions and CORRECT
Answers
How many relations are there on a set |n| ? - CORRECT ANSWER- ✔✔2^(n^2) relations
out degree - CORRECT ANSWER- ✔✔# of things 'a' relates to (# of 1's in the row of the
matrix)
in degree - CORRECT ANSWER- ✔✔# of things that relate to 'a' (# of 1's in the column of
the matrix)
cycle - CORRECT ANSWER- ✔✔a path the begins and ends at the same vertex
reflexive - CORRECT ANSWER- ✔✔-every element is related to itself
-on a digraph, each element will have an arrow pointing to itself
-on a matrix, there will be 1's on the main diagonal
irreflexive - CORRECT ANSWER- ✔✔-no element is related to itself
-on the digraph, no element will have an arrow pointing to itself
-on a matrix, there will be 0's on the main diagonal
symmetric - CORRECT ANSWER- ✔✔- (a, b) ∈ R, then (b, a) ∈ R
-every element in the relation, also has its reverse (if (1,2) is in the relation, (2,1) must also be
in the relation)
-on the digraph, nodes will point at each other (two way streets)
-the original matrix is equal to itself transposed
asymmetric - CORRECT ANSWER- ✔✔- (a, b) ∈ R, then (b, a) ∉ R
, - no element has its reverse (no symmetric pairs)
-on the digraph, all paths are one way
-on the matrix, if Mij = 1, then Mji = 0
-a relation is asymmetric iff it is antisymmetric and irreflexive
-a transitive relation is asymmetric iff it is irreflexive
antisymmetric - CORRECT ANSWER- ✔✔-if (a, b) ∈ R and (b, a) ∉ R, then a=b
-the only symmetric pairs are elements related to themselves
-on the matrix, if i≠j, then Mij = 0 or Mji = 0
transitive - CORRECT ANSWER- ✔✔-(a, b) ∈ R and (b, c) ∈ R, then (a,c) ∈ R
-on the matrix, if Mij = 1 and Mjk = 1, then Mik = 1
-a transitive relation is asymmetric iff it is also irreflexive
equivalence relation - CORRECT ANSWER- ✔✔A relation that is reflexive, symmetric, and
transitive
equivalence class - CORRECT ANSWER- ✔✔an equivalence class is part of an equivalence
relation. If the relation was people are related if they are sitting in the same row, all of the
people in one row would be an equivalence class
closure - CORRECT ANSWER- ✔✔the smallest possible addition to a relation in order to
achieve desired properties (i.e. the smallest amount of elements you could add to a relation to
make it reflexive)
everywhere defined - CORRECT ANSWER- ✔✔-Dom(f) = A
-every element in the domain has at least one corresponding element in the range
surjective - CORRECT ANSWER- ✔✔Ran(f) = B
-for every element in the range, there is at least one corresponding element in the domain
ACTUAL Exam Questions and CORRECT
Answers
How many relations are there on a set |n| ? - CORRECT ANSWER- ✔✔2^(n^2) relations
out degree - CORRECT ANSWER- ✔✔# of things 'a' relates to (# of 1's in the row of the
matrix)
in degree - CORRECT ANSWER- ✔✔# of things that relate to 'a' (# of 1's in the column of
the matrix)
cycle - CORRECT ANSWER- ✔✔a path the begins and ends at the same vertex
reflexive - CORRECT ANSWER- ✔✔-every element is related to itself
-on a digraph, each element will have an arrow pointing to itself
-on a matrix, there will be 1's on the main diagonal
irreflexive - CORRECT ANSWER- ✔✔-no element is related to itself
-on the digraph, no element will have an arrow pointing to itself
-on a matrix, there will be 0's on the main diagonal
symmetric - CORRECT ANSWER- ✔✔- (a, b) ∈ R, then (b, a) ∈ R
-every element in the relation, also has its reverse (if (1,2) is in the relation, (2,1) must also be
in the relation)
-on the digraph, nodes will point at each other (two way streets)
-the original matrix is equal to itself transposed
asymmetric - CORRECT ANSWER- ✔✔- (a, b) ∈ R, then (b, a) ∉ R
, - no element has its reverse (no symmetric pairs)
-on the digraph, all paths are one way
-on the matrix, if Mij = 1, then Mji = 0
-a relation is asymmetric iff it is antisymmetric and irreflexive
-a transitive relation is asymmetric iff it is irreflexive
antisymmetric - CORRECT ANSWER- ✔✔-if (a, b) ∈ R and (b, a) ∉ R, then a=b
-the only symmetric pairs are elements related to themselves
-on the matrix, if i≠j, then Mij = 0 or Mji = 0
transitive - CORRECT ANSWER- ✔✔-(a, b) ∈ R and (b, c) ∈ R, then (a,c) ∈ R
-on the matrix, if Mij = 1 and Mjk = 1, then Mik = 1
-a transitive relation is asymmetric iff it is also irreflexive
equivalence relation - CORRECT ANSWER- ✔✔A relation that is reflexive, symmetric, and
transitive
equivalence class - CORRECT ANSWER- ✔✔an equivalence class is part of an equivalence
relation. If the relation was people are related if they are sitting in the same row, all of the
people in one row would be an equivalence class
closure - CORRECT ANSWER- ✔✔the smallest possible addition to a relation in order to
achieve desired properties (i.e. the smallest amount of elements you could add to a relation to
make it reflexive)
everywhere defined - CORRECT ANSWER- ✔✔-Dom(f) = A
-every element in the domain has at least one corresponding element in the range
surjective - CORRECT ANSWER- ✔✔Ran(f) = B
-for every element in the range, there is at least one corresponding element in the domain
