Let R be an equivalence relation on A = {0,2,3,5} and R = {(5,5), (0, 2), (3, 3), (2, 0), (2, 2), (0, 0)}. There are unique equivalence classes defined by R.

Let R be the relation on the set {a,b,c,d} containing the ordered pairs (a,a), (a,c), (b,b), (b,c), (b,d), (c,b), (c,c), (d,a) and (d,d). Find R².

Determine which constructs of RDFS and OWL can be expressed using monotonic rules. For example, the subclass relation is represented as c(X)→c0(X) (c0 is a subclass of c0).

List all 3-tuples in the relation {(a,b,c)| abc = 6} where a, b and c are all positive integers.

a) Define a relation R on R (the set of all real numbers) as follows: For all x, y E R, xRy → x < y. Is R reflexive, symmetric and transitive?

Consider an equivalence relation R on X={a,b,c}. R= {(a,a), (a,c), (b,b), (c,a), (c,c) } [a]=

Find equivalence classes of following relations if they exist.  a) {(0, 0), (1, 1), (2, 2), (3, 3)} b) {(0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)} c) {(0, 0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 3)} d) {(0, 0), (1, 1), (1, 3), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)} e) {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 2), (3, 3)}

1. Let A = {3, 4, 5} and B = {4, 5, 6} and let S be the "divides" relation. That is, for all (x, y) ≤ A × B, xSy ↔ x|y. State explicitly whic ordered pairs are in S and S-1

Mathematical Logic First-order or predicate logic. Show that the sum relation, {(m,n,p)|p=m+n}, is not definable on (N; ∙). Hint: Consider an automorphism of (N; ∙) that interchanges two primes. Where N is the set of natural numbers. Please be as clear as possible. Show and explain all the steps. Thank you very much.

Let R be an equivalence relation on A = {0,5,7,9} and R = {(0, 9), (0, 5), (7, 7), (9, 5), (5,9), (9, 9), (5, 0), (0, 0), (9, 0), (5,5)}. Answer each of the following questions. (5) = 0 [9] = [0] = [7] =