Discrete Structures Final Exam | Questions and answers | Latest 2024/25

Discrete Structures Final Exam | Questions and answers | Latest 2024/25 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 -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 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 ad d 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 th e 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. 1 2 3 4 5 6 ↓ ↓ ↓ ↓ ↓ ↓ 4 5 6 3 2 1 (1,4,3,6)(2,5) transposition a cycle in a permutation of length 2 ex. in the permutation (1,4,3,6)(2,5), (2,5) is a transposition codomain -All the values that may be output from a mathematical function -not necessarily the same as the range of the function -could be equal to or contain more elements than the range domain all of the input values of a function range -all outputs of a function that correspond with an element from the domain -all values in the range are also in the codomain, b ut not all values in the codomain are necessarily in the range. - could be equal to or contain fewer elements the codomain properties of a relation on the empty set -irreflexive, symmetric, asymmetric, antisymmetric, transitive -basically everything excep t reflexive What is the result of multiplying a permutation by its inverse? the identity (I) ex. (1, 4, 2) (1, 2, 4) = (I)