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Discrete Structures Final Exam
Questions and Answers 100% Solved
How many relations are there on a set |n| ? - ✔✔2^(n^2) relations
out degree - ✔✔# of things 'a' relates to (# of 1's in the row of the matrix)
in degree - ✔✔# of things that relate to 'a' (# of 1's in the column of the
matrix)
cycle - ✔✔a path the begins and ends at the same vertex
reflexive - ✔✔-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 - ✔✔-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 - ✔✔- (a, b) ∈ R, then (b, a) ∈ R
,©JOSHCLAY 2024/2025. YEAR PUBLISHED 2024.
-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 - ✔✔- (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 - ✔✔-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 - ✔✔-(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 - ✔✔A relation that is reflexive, symmetric, and
transitive
, ©JOSHCLAY 2024/2025. YEAR PUBLISHED 2024.
equivalence class - ✔✔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 - ✔✔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 - ✔✔-Dom(f) = A
-every element in the domain has at least one corresponding element in the
range
surjective - ✔✔Ran(f) = B
-for every element in the range, there is at least one corresponding element
in the domain
injective - ✔✔for every element in the range, there is exactly one
corresponding element in the domain.
bijection - ✔✔a function that is both surjective and injective
permutation - ✔✔a bijection from a set to itself
ex.
123456
Discrete Structures Final Exam
Questions and Answers 100% Solved
How many relations are there on a set |n| ? - ✔✔2^(n^2) relations
out degree - ✔✔# of things 'a' relates to (# of 1's in the row of the matrix)
in degree - ✔✔# of things that relate to 'a' (# of 1's in the column of the
matrix)
cycle - ✔✔a path the begins and ends at the same vertex
reflexive - ✔✔-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 - ✔✔-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 - ✔✔- (a, b) ∈ R, then (b, a) ∈ R
,©JOSHCLAY 2024/2025. YEAR PUBLISHED 2024.
-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 - ✔✔- (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 - ✔✔-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 - ✔✔-(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 - ✔✔A relation that is reflexive, symmetric, and
transitive
, ©JOSHCLAY 2024/2025. YEAR PUBLISHED 2024.
equivalence class - ✔✔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 - ✔✔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 - ✔✔-Dom(f) = A
-every element in the domain has at least one corresponding element in the
range
surjective - ✔✔Ran(f) = B
-for every element in the range, there is at least one corresponding element
in the domain
injective - ✔✔for every element in the range, there is exactly one
corresponding element in the domain.
bijection - ✔✔a function that is both surjective and injective
permutation - ✔✔a bijection from a set to itself
ex.
123456
