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| logical operator examples |
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1.
Let p, q, and r be the propositions
p: You have the flu.
q: You miss the final examination.
r: You pass the course.
a.
p ® q
If you have the flu, then you will
miss the final examination.
b.
Ø q «
r
You will not miss the final
examination if and only if you pass the course.
c.
q ® Ør
If you miss the final examination,
then you will fail the course.
d.
p v q V r
You have the flu either you miss
the examination, or you pass the course.
e.
(p ® Ør)
v (q ® Ør)
If you have the flu, then you will
not pass the course, or if you miss the final examination, then you will fail
the course
f.
(p /\ q) V (Øq /\ r)
You have the flu and you miss the
final examination, or you will not miss the final examination and you pass the
course.
2.
Let p, q, and r be the
propositions
p: You get an A on the final exam.
q: You do every exercise in this book.
r: You get an A in this class.
Write these propositions using p, q, and r
and logical connectives.
a.
You get an A in this class,
but you do not do every exercise in this book.
r Ù
Øq
b.
You get an A on the final,
you do every exercise in this book, and you get an A in this class.
p Ù
q Ù r
c.
To get an A in this class,
it is necessary for you to get an A on the final.
r ®
p
d.
You get an A on the final,
but you don't do every exercise in this book; nevertheless, you get an A in
this class.
p Ù
Øq Ù r
e.
Getting an A on the final
and doing every exercise in this book is sufficient for getting an A in this
class.
(p Ù
q) ® r
f.
You will get an A in this
class if and only if you either do every exercise in this book or you get an A
on the final.
r «
(q Ú p)
3.
Construct a truth table for
each of these compound propositions.
a.
p ®(Øq
v r)
|
p
|
q
|
r
|
Ø
q
|
Ø
q Ú r
|
p ®
(Ø q Ú r)
|
|
T
|
T
|
T
|
F
|
T
|
T
|
|
T
|
T
|
F
|
F
|
F
|
F
|
|
T
|
F
|
T
|
T
|
T
|
T
|
|
T
|
F
|
F
|
T
|
T
|
T
|
|
F
|
T
|
T
|
F
|
T
|
T
|
|
F
|
T
|
F
|
F
|
F
|
T
|
|
F
|
F
|
T
|
T
|
T
|
T
|
|
F
|
F
|
F
|
T
|
T
|
T
|
b.
Ø p ®
(q ® r)
|
p
|
q
|
r
|
Ø
p
|
q ®
r
|
Øp
® ( q®r)
|
|
T
|
T
|
T
|
F
|
T
|
T
|
|
T
|
T
|
F
|
F
|
F
|
T
|
|
T
|
F
|
T
|
F
|
T
|
T
|
|
T
|
F
|
F
|
F
|
T
|
T
|
|
F
|
T
|
T
|
T
|
T
|
T
|
|
F
|
T
|
F
|
T
|
F
|
F
|
|
F
|
F
|
T
|
T
|
T
|
T
|
|
F
|
F
|
F
|
T
|
T
|
T
|
c.
(p ® q) v (ØP ®
r)
|
p
|
q
|
r
|
Ø
p
|
p ®
q
|
Øp
® r
|
(p ®
q) Ú (Øp ® r)
|
|
T
|
T
|
T
|
F
|
T
|
T
|
T
|
|
T
|
T
|
F
|
F
|
T
|
T
|
T
|
|
T
|
F
|
T
|
F
|
F
|
T
|
T
|
|
T
|
F
|
F
|
F
|
F
|
T
|
T
|
|
F
|
T
|
T
|
T
|
T
|
T
|
T
|
|
F
|
T
|
F
|
T
|
T
|
F
|
T
|
|
F
|
F
|
T
|
T
|
T
|
T
|
T
|
|
F
|
F
|
F
|
T
|
T
|
F
|
T
|
d.
(p ® q) Ù ( Ø
p ® r)
|
p
|
q
|
r
|
Ø
p
|
p ®
q
|
Øp
® r
|
(p ®
q) Ù (Øp ® r)
|
|
T
|
T
|
T
|
F
|
T
|
T
|
T
|
|
T
|
T
|
F
|
F
|
T
|
T
|
T
|
|
T
|
F
|
T
|
F
|
F
|
T
|
F
|
|
T
|
F
|
F
|
F
|
F
|
T
|
F
|
|
F
|
T
|
T
|
T
|
T
|
T
|
T
|
|
F
|
T
|
F
|
T
|
T
|
F
|
F
|
|
F
|
F
|
T
|
T
|
T
|
T
|
T
|
|
F
|
F
|
F
|
T
|
T
|
F
|
F
|
e.
(p « q)v( Øq «
r)
|
p
|
q
|
r
|
Ø
q
|
p «
q
|
Øq
« r
|
(p «
q) Ú (Øq « r)
|
|
T
|
T
|
T
|
F
|
T
|
F
|
T
|
|
T
|
T
|
F
|
F
|
T
|
T
|
T
|
|
T
|
F
|
T
|
T
|
F
|
T
|
T
|
|
T
|
F
|
F
|
T
|
F
|
F
|
F
|
|
F
|
T
|
T
|
F
|
F
|
F
|
F
|
|
F
|
T
|
F
|
F
|
F
|
T
|
T
|
|
F
|
F
|
T
|
T
|
T
|
T
|
T
|
|
F
|
F
|
F
|
T
|
T
|
F
|
T
|
f.
( Ø p «
Øq) « (q«r)
|
p
|
q
|
r
|
Øp
|
Ø
q
|
Øp
« Øq
|
q « r
|
(Øp
« Øq) « (q « r)
|
|
T
|
T
|
T
|
F
|
F
|
T
|
T
|
T
|
|
T
|
T
|
F
|
F
|
F
|
T
|
F
|
F
|
|
T
|
F
|
T
|
F
|
T
|
F
|
F
|
T
|
|
T
|
F
|
F
|
F
|
T
|
F
|
T
|
F
|
|
F
|
T
|
T
|
T
|
F
|
F
|
T
|
F
|
|
F
|
T
|
F
|
T
|
F
|
F
|
F
|
T
|
|
F
|
F
|
T
|
T
|
T
|
T
|
F
|
F
|
|
F
|
F
|
F
|
T
|
T
|
T
|
T
|
T
|
4.
Evaluate each of these expressions.
a.
I 1000 Ù
(0 1011 Ú I
1011)
0 1011 Ú
I 1011 = 11011
I 1000 Ù 11011 = 11000
b.
(0 IIII Ù I 0101) Ú
0 1000
0 IIII Ù I 0101 = 00101
00101 Ú
0 1000 = 01101
c.
(01010 Å 11011) Å 0 1000
01010 Å 11011 = 10001
10001 Å 0 1000 = 11001
d.
(11011 Ú 0 1010) Ù (10001 Ú 11011)
11011 Ú 0 1010 = 11011
10001 Ú 11011 = 11011
11011 Ù 11011 = 11011
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