COT 3100 Homework # 4
Spring 2000
Assigned: 3/7/00
Due: 3/30,31 in recitation
1)
Define a relation T
⊆
N x N such that T = {(a,b) a
∈
A
∧
b
∈
A
∧
a – b = 2c+1 for some
integer c}.
(N is the set of nonnegative integers.)
a) Prove that this relation is not reflexive.
b) Prove that this relation is symmetric.
c) Define the term antitransitive as the following:
Given a set A and a relation R,
if for all a,b,c
∈
A, (aRb
∧
bRc
∧
cRa)
⇒
(a = b
∨
b = c)
Prove that the relation T is antitransitive.
2)
Let g: A
→
A be a bijection. For n
≥
2, define g
n
= g
°
g
°
...
°
g, where g is composed
with itself n times. Prove that for n
≥
2, that g
n
is a bijection from A to A as well, and
show that (g
n
)
1
= (g
1
)
n
.
3)
Let f : A
→
B and g: B
→
C denote two functions. If both f and g are injective, prove
that the composition g
°
f: A
→
C is an injection as well.
4)
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 Spring '09
 Equivalence relation, Binary relation, Transitive relation, Preorder, Transitive closure

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