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CS 70
Discrete Mathematics for CS
Spring 2000
Wagner
Midterm 1
You have 1 hour and 20 minutes. The exam is openbook, opennotes.
100 points total (five questions of varying credit as shown).
You should be able to finish all the questions, so avoid spending too long on any one question.
.
Write your answers in blue books. Check you haven’t skipped any by accident. Hand them all in. Panic not.
1. (20 pts.)
Satisfiability and all that
For each of the following Boolean expressions, decide if it is (i) valid (ii) satisfiable (iii) unsatisfiable.
(Give
all
applicable properties, with justifications.)
(a) (5)
A
∧¬
A
∧¬
B
(b) (10)
(
A
=
⇒
B
)
∧
(
B
=
⇒
C
)
∧
(
C
=
⇒ ¬
A
)
(c) (5)
(
A
=
⇒
B
)
∨
(
B
=
⇒
A
)
2. (25 pts.)
Induction
Given a binary tree
t
, the depth
depth
(
l
,
t
)
of a leaf
l
in
t
is defined to be the length of the path from
l
to the root of
t
. (Hence, if
t
is just an atom
l
, then
depth
(
l
,
t
) =
0.)
(a) (3) In the tree
t
below, what are the depths of the 3 leaves
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 Fall '08
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