Problem 1: Let X = {1, 2, 3, 4} and Y = {3, 4, 5, 6}. Define a relation Q from X to Y by the rule: x Q y if and only if x >= y.
a. Which of the following statements are true of the relation Q? .
Statement
True or False?
Reason
2 Q 4
4 Q 3
(4,4) is in Q
(7,3) is in Q
b. Write Q as a set of ordered pairs. Be sure to use correct notation for both sets and ordered pairs.
Important hint: Check your answers to (a) and (b) for consistency with one another!
In Problems 2 and 3, you are given a binary relation on a set. Determine whether the relation is (a) reflexive, (b) symmetric, (c) transitive. If your answer to any of the questions about the relation is No, give a specific counterexample. (There is no need to give more than one counterexample to justify a "No".)
Problem 2:
Set: {0, 1, 2, 3}
Relation: {(0,0), (0,1), (1,1), (1,2), (2,2), (2,3), (3,3)}
Reflexive?
Symmetric?
Transitive?
Problem 3:
Set: the set Z of integers.
Relation O: m O n means m - n is odd.
Reflexive?
Symmetric?
Transitive?
Hint: Experiment with a couple dozen examples first.
Problem 4: Draw a directed graph to represent the relation R on A, where A = {1, 2, 3, 4, 5} and R = {(1,2), (1,3), (2,2), (2,4), (4,5)}.
Problem 5: Let A = {a, b, c, d, e, f} be partitioned into subsets {a, b}, {c, e, f}, and {d}. Illustrate the corresponding equivalence relation by a directed graph.
Problem 6. Define a relation L on R x R (i.e the Euclidean plane with the usual Cartesian coordinates) by the rule (x, y) L (z, w) if and only if x - y = z - w. Show that L is an equivalence relation. Then describe the equivalence classes.