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MODULE ONE PROBLEM SET
This document is proprietary to Southern New Hampshire University. It and
the problems within may not be posted on any non-SNHU website.
Jacqueline Amoah
1
Directions: Type your solutions into this document and be sure to show all steps
for arriving at your solution. Just giving a final number may not receive full credit.
P
ROBLEM
1
In the following question, the domain of
discourse
is a set of male patients in a
clinical study. Define the following predicates:
P (x) : x was given the placebo
D(x) : x was given the medication
M (x) : x had migraines
Translate
i
each
i
of
i
the
i
following
i
statements
i
into
i
a
i
logical
i
expression.
i
Then
i
negate
i
the
i
expression
i
by
i
adding
i
a
i
negation
i
operation
i
to
i
the
i
beginning
i
of
i
the
i
expression.
i
Apply
i
De
i
Morgan’s
i
law
i
until
i
each
i
negation
i
operation
i
applies
i
directly
i
to
i
a
i
predicate
i
and
i
then
i
translate
i
the
i
logical
i
expression
i
back
i
into
i
English.
Sample
i
question:
i
Some
i
patient
i
was
i
given
i
the
i
placebo
i
and
i
the
i
medication.
x (P (x)
D(x))
Negation:
¬∃
x (P (x)
D(x))
Applying
De
M
o
r
ga
n
’s
law:
x
(
¬
P
(
x
)
¬
D
(
x
))
English: Every patient was either not given the placebo or not given the
medication (or both).
¬ ¬
¬∀
¬
¬¬
−→ ¬
¬
−→
¬ ¬
¬∃
(a)
Every patient was given the medication or the placebo or both.
x (D(x)
P (x))
Negation:
x (D(x) P (x))
Applying De Morgan’s law: x( D(x)
P (x))
English:
i
There
i
is
i
a
i
patient
i
who
i
was
i
not
i
given
i
the
i
medication
i
and
i
not
i
given
i
the
i
placebo.
(b)
Every patient who took the placebo had migraines. (Hint: you will need to
apply the conditional identity, p q ¬p q.)
x (P (x)
M (x))
Negation:
x (P (x) M (x))
Applying Conditional identity: (P (x)
M (x)) ( P (x) M (x))
Applying Double Negation;
P (x) P (x)
Applying De Morgan’s law: x (P (x)
M (x))
English:
i
Some
i
patient
i
took
i
the
i
placebo
i
and
i
did
i
not
i
have
i
migraines.
(c)
There is a patient who had migraines and was given the placebo.
x (M (x)
P (x))
Negation:
x (M (x)
P (x))
Applying De Morgan’s law: x ( M (x)
P (x))
English:
i
Every
i
patient
i
did
i
not
i
have
i
migraines
i
or
i
did
i
not
i
take
i
the
i
placebo.
P
ROBLEM
2
Use
i
De
i
Morgan’s
i
law
i
for
i
quantified
i
statements
i
and
i
the
i
laws
i
of
i
propositional
i
logic
i
to
i
show
i
the
i
following
i
equivalences:
(a)
¬∀
x (P (x)
¬
Q(x))
x (
¬
P (x)
Q(x))
¬∀
x (P (x)
¬
Q(x))
Applying De Morgan’s Law:
x (
¬
(P (x)
¬¬
Q(x))
Applying Double Negation:
x (
¬
P (x)
Q(x))
Therefore
x (
¬
P (x)
Q(x))
x (
¬
P (x)
Q(x))
(b)
¬∀
x (
¬
P (x)
Q(x))
x (
¬
P (x)
¬
Q(x))
¬∀
x (
¬
P (x)
−→
Q(x))
Applying Conditional identity: (
¬¬
P (x)
Q(x))
Applying Double Negation: (P (x)
Q(x))
Applying De Morgan’s Law:
x (
¬
P (x)
¬
Q(x))
Therefore
x (
¬
P (x)
¬
Q(x))
x (
¬
P (x)
¬
Q(x))
(c)
¬∃
x
¬
P
(
x
)
(
Q
(
x
)
¬
R
(
x
))
x
P
(
x
)
(
¬
Q
(
x
)
R
(
x
))
¬∃
x (
¬
P (x)
(Q(x)
¬
R(x)))
Applying
De
M
organ’s
La
w:
x
(
P
(
x
)
¬
(
Q
(
x
)
¬
R
(
x
)))
x
(
P
(
x
)
(
¬
Q
(
x
)
¬¬
R
(
x
)))
Applying
Double
Negation:
x
(
P
(
x
)
(
¬
Q
(
x
)
R
(
x
)))
Therefore
x
(
P
(
x
)
(
¬
Q
(
x
)
R
(
x
)))
x
(
P
(
x
)
(
¬
Q
(
x
)
R
(
x
)))
P
ROBLEM
3
The
i
domain
i
of
i discourse
i
for
i
this
i
problem
i
is
i
a
i
group
i
of
i
three
i
people
i
who
i
are
i
working
i
on
i
a
i
project.
i i
To
i
make
i
notation
i
easier,
i
the
i
people
i
are
i
numbered
i
1,
i
2,
i
3.
i
The
i
predicate
i
M
i
(x,
i i
y)
i
indicates
i
whether
i
x
i
has
i
sent
i
an
i
email
i
to
i
y,
i i
so
i
M
i
(2,
i i
3)
i
is
i
read
i
“Person
i
2
i
has
i
sent
i
an
i
email
i
to
i
person
i
3.”
i
The
i
table
i
below
i
shows
i
the
i
value
i
of
i
the
i
predicate
i
M
i
(x,
i
y)
i
for
i
each
i
(x,
i i
y)
i
pair.
i i
The
i
truth
i
value
i
in
i
row
i
x
i
and
i
column
i
y
i
gives
i
the
i
truth
i
value
i
for
i
M
i
(x,
i
y).
M
1
2
3
1
T
T
T
2
T
F
T
3
T
T
F
Determine if the quantified statement is true or false. Justify your an-
swer.
(a)
x
y
(
x
y
)
M
(
x,
y
))
False:i Thisi isi becausei accordingi toi thei firsti rowi ofi thei truthi table,i
Person1i senti ani emaili toi everyonei includingi themselves.
(b)
x
y
¬
M
(
x,
y
)
False:
i
Everyone
i
received
i
an
i
email
i
from
i
someone
i
according
i
to
i
the
i
truth
i
table.
(c)
x y M (x, y)
True:i Someonei senti ani emaili toi everyi one.i Person1i senti ani emaili toi
every-i onei includingi themselves.
P
ROBLEM
4
Translate
i
each
i
of
i
the
i
following
i
English
i
statements
i
into
i
logical
i
expressions.
i
The
i
domain
i
of
i discourse
i
is
i
the
i
set
i
of
i
all
i
real
i
numbers.
(a)
The
i
reciprocal
i
of
i
every
i
positive
i
number
i
less
i
than
i
one
i
is
i
greater
i
than
i
one.
x
((
x
>
0)
((
x
<
1)
(1
/x
>
1)))
(b)
There
i
is
i
no
i
smallest
i
number.
x
(
x
>
inf
inity
)
(c)
Every
i
number
i
other
i
than
i
0
i
has
i
a
i
multiplicative
i
inverse.
x
(
x
!
=
0
1
/x
)
P
ROBLEM
5
×
The sets A, B, and C are defined as follows:
A = tall, grande, venti
B = foam, no
foam
C = non
fat, whole
Use
i
the
i
definitions
i
for
i
A,
i
B,
i
and
i
C
i
to
i
answer
i
the
i
questions.
i
Express
i
the
i
elements
i
using
i
n-tuple
i
notation,
i
not
i
string
i
notation.
(a)
Write
i
an
i
element
i
from
i
the
i
set
i
A
i
×
i
B
i
×
i
C.
i
(tall,
i
no-foam,
i
whole)
(b)
Write
i
an
i
element
i
from
i
the
i
set
i
B
i
×
i
A
i
×
i
C.
i
(foam,
i
venti,
i
non-fat)
(c)
Write
i
the
i
set
i
B
i
×
i
C
i
using
i
roster
i
notation.
B
C
i
=
i
(foam,
i
non-fat),
i
(foam,
i
whole),
i
(no-foam,
i
non-fat),
i
(no-
foam,
i
whole)
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