Homework Problems
What is the probability that a random relation from set A = {a, b, 0, d} [0 sel B = {1, 2, 3, 4, 8} is a one-to-one function?
Consider the set A = {1, 2, 3, 4}. On the cartesian product A x A, we define the relation R by (x, y) R (z, y) if and only if x = z. Show that R is an equivalence relation and illustrate the different equivalence classes in figure_
Consider the following set S = {(a, b) | a, b ∈ Z} where Z denotes the integers. Show that the relation (a, b) R (e, d) if and only if ad = be on S is an equivalence relation. Give the equivalence class [(1, 2)]: What can an equivalence class be associated with?
Consider set A = {1, 2, 3, 4, 5, 6}. Define an equivalence relation R on A which realizes {1, 3, 5} and {2, 4, 6} as the partition of A.
Two Hasse diagrams are shown below: Unfortunately, the labels at the vertices are missing. The diagram on the right represents the divides relation, i.e. (x, y) ∈ R if and only if x divides y, and the set is some subset of the positive integers. For the diagram on the left, the relation is the inclusion relation, i.e. (X, Y) ∈ R if and only if X is a subset of Y, and the set is a subset of the power set of a finite set.
Figure [: Hasse Diagram