Question

Assume the relation R over the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} obtained by combining adjacent (neighbouring) digit pairs of your ID, keeping the same order of digits. For example, if my ID is 12345678, then my relation R would be {(1,2),(3,4),(5,6),(7,8)}. If your ID has an odd number of digits, then repeat the last digit my ID =401902258. Determine if the relation is Reflexive, symmetric, antisymmetric, or transitive (Use a table). Represent your R by a graph, and specify what type of graph can be used to represent this relation. Generate a matrix (MR) for your relation R. Generate the matrix for the composition of R and R (MR or) for your relation.

Answer 1

Given the ID 401902258, we can determine that the relation R is not reflexive, not symmetric, antisymmetric, and potentially transitive. A directed graph can represent the relation. The adjacency matrix and composition matrix are both 10x10 matrices indicating the presence or absence of relations and compositions respectively.

Step-by-step explanation:

To determine if the given relation is Reflexive, Symmetric, Antisymmetric, or Transitive, we'll consider each property based on the ID number provided, 401902258. Our relation R consists of the digit pairs (4,0), (1,9), (0,2), (9,0), (2,2), (2,5), (5,8), and (8,8).

• Reflexive: A relation is reflexive if every element is related to itself. Since not all elements in our set are paired with themselves in R (e.g., we do not have (1,1) or (3,3)), R is not reflexive.
• Symmetric: A relation is symmetric if for every pair (a,b) in R, the pair (b,a) is also in R. Since we have (4,0) but not (0,4), and (1,9) but not (9,1), R is not symmetric.
• Antisymmetric: A relation is antisymmetric if for every pair (a,b) in R, whenever (a,b) and (b,a) are both in R, then a must equal b. Since we do not have any such cases where both (a,b) and (b,a) are in R for distinct a and b, R is antisymmetric.
• Transitive: A relation is transitive if whenever (a,b) and (b,c) are in R, then (a,c) must also be in R. Given that we do not have sufficent pairs to violate this condition, R could potentially be transitive, but we cannot confirm without reviewing all possible combinations.

The relation R can be represented by a directed graph where each ordered pair is a directed edge from the first number to the second.

The adjacency matrix (MR) for the relation R is a 10x10 matrix where the row represents the first element of the ordered pair, and the column represents the second. The matrix will have '1' if the relation between the row and column elements exists in R, and '0' otherwise.

The matrix for the composition of R and R (R comp R), denoted as MR o R, is also a 10x10 matrix constructed by taking the logical AND of the products of corresponding elements of the rows of MR and the columns of MR.