CS 173: Discrete Mathematical Structures, Fall 2008
Homework 10
Due at class on Friday, November 21, 2008
1. [15 points] Relations and Closures
Here is the graph representation of a relation Ron the set X={A, B , C, D, E}.
A
B
C
D
E
(a) Write Ras a list of pairs.
(b) Write Ras a zero-one matrix.
(c) Is Rsymmetric? antisymmetric? Explain briefly why or why not.
(d) Draw the diagram for the transitive closure of R.
2. [10 points] Equivalence relation proof
Define a relation Ron the integers such that pRq if and only if 5|2p+ 3q.
(a) Prove that Ris an equivalence relation.
(b) List the equivalence classes of Rand describe clearly what’s in each of them.
3. [10 points] Well-defined operations
Define an equivalence relation ∼on R3− {(0,0,0)}(triples of real numbers) by (x, y , z)∼
(a, b, c)if and only if there is a positive real number λsuch that (x,y , z) = λ(a, b, c). The
equivalence classes of ∼are the rays from the origin. Let’s use Vto name the set of equiva-
lence classes, i.e. the set of rays.
Here are two operations defined on V. Prove that each is well-defined. That is, to compute
the value of the operation on two input rays, you need to pick a representative point from
each ray. Show that the output equivalence class does not depend on which representatives
you select.
(a) The dot product operation: [(x, y , z)] ·[(p, q, r)] = [(xp, y q, zr)]
(b) The cross product operation: [(x, y , z)] ×[(p, q, r)] = [(yr −zq, z p −xr, xq −yp)]
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