MATH 327, HOMEWORK #3
Problem 1. Suppose A={โ3,โ2,โ1,0,1,2,3}.Define a relation ๎
by
a๎bmeans (aโ1)2โค(bโ1)2
Show that this is reflexive and transitive, but is not antisymmetric.
Problem 2. Consider the following parital order on the set ZรZ:
(a, b)๎(x, y) means a < x or (a=xand bโคy)
With this partial order, answer the following.
(a) Find the least upper bound of (โ2,3) and (โ2,5) or explain
why it does not exist.
(b) Find the least upper bound of (1,3) and (4,5) or explain why
it does not exist.
Problem 3. Consider the following parital order on the set ZรZ:
(a, b)๎(x, y) means a=xand bโคy
With this partial order, answer the following.
(a) Find the least upper bound of (โ2,3) and (โ2,5) or explain
why it does not exist.
(b) Find the least upper bound of (1,3) and (4,5) or explain why
it does not exist.
Problem 4. Consider all the possbible functions from
A={1,2,3,4,5}
to
B={2,4,6,8,10}
that on 1 take value 2,and on 2 take value 4,and on 3 take value 6.
That is, we require
17โ 2,27โ 4,and 3 7โ 6,
(a) How many such functions are there?
(b) How many are one-to-one?
(c) How many are onto?
(d) How many have have range equal to {2,4,6}?
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