MAD 2104
Dec 8, 2011
Final Exam
Prof. S. Hudson
1) [10 pts] How many bit strings of length 7 contain at least ﬁve 1’s ?
2) [10 pts] Prove ONE, in the usual paragraph style, including the phrase
x
∈
more than
once. Avoid shortcuts, such as quoting stronger theorems about sets (see me, if in doubt
about this).
A
∪
B
⊆
A
∩
B
If
A
⊆
B
, then
B
⊆
A
.
3) [5 points each] Let
A
=
{
0
,
1
,
2
}
.
a) Give an example of a relation
R
on
A
, that is reﬂexive but not symmetric. You
may describe
R
in words, or by a clearlylabeled matrix, or (di)graph.
b) Give an example of a diﬀerent relation
R
on
A
, that is transitive but not an
equivalence relation. If either part of this problem is impossible, explain why.
4) [10 points] Find a Boolean expression in sumofminterm form (eg DNF) for the function,
F
(
x,y,z
) = 1 if and only if
x
+
y
= 0.
5) [15 points] Answer True or False:
If
f
:
A
→
A
is onto, it must also be onetoone.
The wheel graph
W
n
for
n
≥
3 is never bipartite.
The complete graph
K
n
for
n
≥
4 always has an even number of edges.
The number of diﬀerent Boolean functions of degree
n
is always at least 4
n
(for
n
≥
1).
The terms of the Fibonacci sequence alternate between odd and even forever.
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 Fall '08
 STAFF
 Boolean Algebra, Equivalence relation, Binary relation, Isomorphism

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