Material for Quiz Eight

10.1:  13-15, 21-27, 32-33
10.2:  2-36 even
10.3:  1-9, 14-25

Quiz Eight:
Average score: 8.11

1.  Which of the following is a partition of {a,b,c,d}?  For the partitions,
draw the digraph (directed graph) of the relation induced by the partition.
 a)  {a,b},{a,c},{c,d} is NOT a partition, because the subsets are not
	disjoint.
 b)  {a,b,c} is NOT a partition because it does not span all of {a,b,c,d}.
 c)  {a},{b,c,d} IS a partition.  The graph would be the point "a" looped
	to itself, and then the points "b" "c" and "d" all connected.

2.  For n and m positive integers, define the relation R as nRm iff
	m mod(n) = 0.  Is R reflexive?  Symmetric?  Transitive?  Prove
	or give counterexamples.
 a)  R is reflexive.  By the Q-R theorem, m =1*m + 0.
 b)  R is not symmetric.  6R3, since 6 mod(3) = 0, but 3 mod(6) = 3, 
	so 3/R6 .
 c)  R is transitive.  Suppose mRn and nRp.  By Q-R this means m = kn,
	and n = hp .  Thus m = k(hp) = (kh)p, and by closure, mRp .

3.  Ley x,y be real numbers and let C be the "circle" relation, i.e.
	xCy iff x^2 + y^2 = 1.  Is C reflexive?  Symmetric?  Transitive?
	Prove or give counterexamples.
 a)  C is not reflexive.  1+1=2 .
 b)  C is symmetric, since addition is symmetric.  That is, x^2 + y^2 = 
	y^2 + x^2 .
 c)  C is not transitive.  1C0, and by symmetry 0C1, but as we saw in (a),
	1/C1 .