CMPS 211 Homework Assignment 01 Solution, Fall 2014
American University of Beirut · Discrete Structures · 3 pages · with solution
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CMPS
211
-‐
Fall
2014
1
Assignment
1
Solution
Exercise
1:
a) It
is
not
a
proposition
b) It
is
a
proposition,
false:
there
are
black
flies
in
Maine.
c) It
is
not
a
proposition
d) If
z
belongs
to
R,
this
is
a
true
proposition
If
z
belongs
to
C,
this
is
not
a
proposition
e) It
is
not
a
proposition
f) It
is
a
proposition
True:
my
money
is
spent
on
kittens
False:
my
money
is
not
spent
on
kittens
Exercise
2:
a) If
I
don’t
run
5km
each
Sunday,
then
I’m
not
athletic.
b) I
don’t
run
5km
each
Sunday
and
I’m
not
athletic.
c) I’m
either
not
athletic
or
I
run
5km
each
Sunday
and
I
am
athletic.
Exercise
3:
a) ¬r
→
¬q
b) r
∧
q
∧
p
c) p
∧
¬r
∧
q
d) p
→
r
e) r
→
q
f) (¬p
→
¬q)
∧
(¬q
→
¬r)
Exercise
4:
a) True
b) True
c) False
d) True
Exercise
5:
a) Inclusive
or
b) Inclusive
or
c) Exclusive
or
d) Exclusive
or
CMPS
211
-‐
Fall
2014
2
Exercise
6:
a) Converse:
if
it
is
mind
bogglingly
complex,
it
looks
easy.
Contrapositive:
if
it
is
not
mind
bogglingly
complex,
it
doesn’t
look
easy.
Inverse:
if
it
doesn’t
look
easy,
it
is
not
mind
bogglingly
complex.
b) Converse:
it
is
a
sunny
summer
day
whenever
I
go
to
the
beach.
Contrapositive:
it
is
not
a
sunny
summer
day
whenever
I
don’t
go
to
the
beach.
Inverse:
I
don’t
go
to
the
beach
whenever
it
is
not
a
sunny
summer
day.
c) Converse:
when
I
sleep
until
noon,
it
is
necessary
that
I
stay
up
late.
Contrapositive:
when
I
don’t
sleep
until
noon,
it
is
necessary
that
I
don’t
stay
up
late.
Inverse:
when
I
don’t
stay
up
late,
it
is
necessary
that
I
don’t
sleep
until
noon.
Exercise
7:
a) ¬p
→
¬q
p
q
¬p
¬q
¬p
→
¬q
T
T
F
F
T
T
F
F
T
T
F
T
T
F
F
F
F
T
T
T
b) p
↔
¬p
p
¬p
p
↔
¬p
T
F
F
T
F
F
F
T
F
F
T
F
c) (¬p
∨
¬q)
→
(
p
∧
¬q)
p
q
¬p
¬q
¬p
∨
¬q
p
∧
¬q
(¬p
∨
¬q)
→
(
p
∧
¬q)
T
T
F
F
F
F
T
T
F
F
T
T
T
T
F
T
T
F
T
F
F
F
F
T
T
T
F
F
d) ¬p
⊕
(p
∨
¬q)
p
q
¬p
¬q
p
∨
¬q
¬p
⊕
(p
∨
¬q)
T
T
F
F
T
T
T
F
F
T
T
T
F
T
T
F
F
T
F
F
T
T
T
F
CMPS
211
-‐
Fall
2014
3
e) (¬p
↔
¬q)
↔
(q
→
p)
p
q
¬p
¬q
¬p
↔
¬q
q
→
p
(¬p
↔
¬q)
↔
(q
→
p)
T
T
F
F
T
T
T
T
F
F
T
F
T
F
F
T
T
F
F
F
T
F
F
T
T
T
T
T
f) (p
↔
q)
⊕
(p
↔
¬q)
p
q
¬q
p
↔
q
p
↔
¬q
(p
↔
q)
⊕
(p
↔
¬q)
T
T
F
T
F
T
T
F
T
F
T
T
F
T
F
F
T
T
F
F
T
T
F
T
Exercise
8:
a) If
x
-‐
2
<
0
then
x
:=
(x+1)/2
-‐1
<
0
=>
True.
So
x=2/2
=
1
b) If
(2x
+2
=3)
OR
(3x
+1
=
3)
then
x
:=
x*2
(2
+
2
≠
3)
and
(4
≠
3)
=>
False
or
False
=>
False.
So
x
=
1
c) if
(x
+3
=
2)
AND
(3x
+4
=
1)
then
x
:=
x-‐2
(4
≠
2)
and
(7≠1)
=>
False
and
False
=>
False.
So
x
=
1
d) if
(x
>
x)
XOR
(x
<
x)
then
x
:=
x+2
(1=1)
=>
False
xor
False
=>
False.
So
x
=
1
e) if
x
<
2
then
x
:=
x
mod
2
(1
<
2)
=>
True.
So
x
=
1
mod
2
=
1
Exercise
9:
a) Joyce
is
not
angry:
truth
value
=
1
-‐
0.9
=
0.1
Martha
is
not
hungry:
truth
value
=
1
-‐
0.3
=
0.7
b) Joyce
is
angry
and
Martha
is
hungry:
truth
value
=
0.3
Neither
Joyce
is
angry
nor
Martha
is
hungry:
truth
value
=
0.1
c) Joyce
is
angry,
or
Martha
is
hungry:
truth
value
=
0.9
Joyce
is
not
angry,
or
Martha
is
not
hungry:
truth
value
=
0.7