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Discrete Structures Final Exam Questions and Answers 100% Solved
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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 ...

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  • October 22, 2024
  • 17
  • 2024/2025
  • Exam (elaborations)
  • Questions & answers

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  • Discrete
  • Discrete

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©JOSHCLAY 2024/2025. YEAR PUBLISHED 2024.

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

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