Discussion 10
Spring 2015
week of March 30
1. Relations—Look at last week’s notes
2. Closure definitions for the relations—For example, the symmetric closure of a relation R is
the smallest symmetric relation that contains R.
a. Let R be the relation on {1, 2, 3, 4} defined by R = {(1, 2), (1, 4), (2, 3), (3, 1), (4, 2)}.
Then, find the following:
1) reflexive closure of R
2) symmetric closure of R
3) transitive closure of R
b. Find the smallest equivalence relation on {a, b, c, d, e} that contains the relation
{(a, b), (a, c), (b, d), (d, e)}.
c. Find the smallest partial order on {a, b, c, d, e} that contains the relation
{(a, b), (a, c), (b, d), (d, e)}.
d. Draw the Hasse diagram for the divides relation on set A = {2, 3, 4, 6, 8, 9, 12, 18}.
e. Prove that the divides relation on the positive integers is a partial order.
f. Prove that if R is a partial order on set A then R-1 is also a partial order on A. Be sure to
prove all three properties.
g. Prove that if R is a nonempty, reflexive and transitive relation on S then, then R  R-1 is
an equivalence relation. Here’s an example of the proof of the symmetric property.
Let (a, b)  R  R-1.
Then (a, b)  R and (a, b)  R-1.
Then, (b, a)  R-1 and (b, a)  R
Thus, (b, a)  R  R-1
Therefore R  R-1 is symmetric
Defn. of intersection
Defn of inverse
Defn of intersection
Defn. of symmetric (a, b)  R  (b, a)  R
h. Let A = {1, 2, 3, 4, 6, 8, 12, 18, 24}. Construct a Hasse diagram for R = {(a, b) | a | b}
i. Using the Hasse diagram below, list all ordered pairs that contain c, and explain how to
determine the total number of ordered pairs in this partial order.