CS1502, Spring 2009

These are good exercises to help you prepare for Exam 2.  However,
this document does not cover all types of questions on the exam.  For
example, there are several questions here that are simply yes and no
questions; on the exam, you will often need to provide more than a
simple yes or no answer.  As stated elsewhere, the questions on the
exam will be related to a question on the homeowork, and example
covered in class, or a "you try it" exercise.

Part I:====================
Which of the following are true, and which are false?
(Read the first one as "A logically follows from A^B" or "A is
entailed by A^B".  Though the book does not use this terminology
often, it is common terminology).

1.   A ^ B |=  A
2.   A     |=  A ^ B
3.   A v B |=  B
4.   B     |=  A v B
5.   ~(P ^ A) |= ~P v ~A
6.   ~P v ~A  |= ~(P ^ A)
7.   P |= Q
8.   Q |= P
9.   ~P |= P
10.   P |= ~P
11.   P ^ (~P v Q) |= Q
12.   Q |= P ^ (~P v Q)
13.  all x P(x) |= ~exists x ~P(x)
14.  ~exists x ~P(x) |= all x P(x)

Which of the following are logical truths?
Which are logically satisfiable but not logical truths?
Which are not logically satisfiable?
(Note:  these involve the same sentences as above)

16.  (A ^ B) --> A
17.  A --> (A ^ B)
18.  (A v B) -->  B
19.  B  -->  (A v B)
20.  ~(P ^ A) -->  (~P v ~A)
21.  (~P v ~A)  --> ~(P ^ A)
22.  P --> Q
23.  Q --> P
24.  ~P --> P
25.  P --> ~P
26.  (P ^ (~P v Q)) --> Q
27.  Q --> (P ^ (~P v Q))
28.  all x P(x) --> ~exists x ~P(x)
29.  ~exists x ~P(x) --> all x P(x)
30.  (P ^ (~P v Q)) --> Q

31.  (A ^ B) <--> A
32.  A <--> (A ^ B)
33.  (A v B) <-->  B
34.  B  <-->  (A v B)
35.  ~(P ^ A) <-->  (~P v ~A)
36.  (~P v ~A)  <--> ~(P ^ A)
37.  P <--> Q
38.  Q <--> P
39.  ~P <--> P
40.  P <--> ~P
41.  (P ^ (~P v Q)) <--> Q
42.  Q <--> (P ^ (~P v Q))
43.  all x P(x) <--> ~exists x ~P(x)
44.  ~exists x ~P(x) <--> all x P(x)
45.  (P ^ (~P v Q)) <--> Q
 
Which of the following pairs are logically equivalent?
(Note1:  these involve the same sentences as above)

46.  (A ^ B), A 
47.  A, (A ^ B)
48.  (A v B),  B
49.  B ,  (A v B)
50.  ~(P ^ A),  (~P v ~A)
51.  (~P v ~A) , ~(P ^ A)
52.  P, Q
53.  Q, P
54.  ~P, P
55.  P, ~P
56.  (P ^ (~P v Q)), Q
57.  Q, (P ^ (~P v Q))
58.  all x P(x), ~exists x ~P(x)
59.  ~exists x ~P(x), all x P(x)
60.  (P ^ (~P v Q)), Q

Which of the following are logically valid arguments?

60.5

    P ^ ~P
    ------
    Q v R


61.
  (A ^ B)
  ---
   A 

62.
  A
 ---
 (A ^ B)

63.
  (A v B)
  ---
  B

64.
  B 
  ---
 (A v B)

65.
  ~(P ^ A)
   ---
  (~P v ~A)

66.
  (~P v ~A) 
   ---
   ~(P ^ A)

67.
  P
  ---
  Q

68.
  Q
  ---
  P

69.
  ~P
  --
  P

70.
  P
  ---
  ~P

71.
  (P ^ (~P v Q))
  ---
  Q

72.
  Q
  ---
 (P ^ (~P v Q))

73.
  all x P(x)
  ---
 ~exists x ~P(x)

74.
  ~exists x ~P(x)
  ---
  all x P(x)

75.
  (P ^ (~P v Q))
  ---
   Q

Part II:=============================
For each of the following, state whether it is a valid argument.
If it is not valid, show that it is not valid.

all x (Student(x) --> Smart(x))
all x Student(x)
---
all x Smart(x)

--------------------------
all x Student(x)
all x Smart(x)
---
all x (Student(x) ^ Smart(x))

--------------------------
exists x (Student(x) --> Smart(x))
exists x Student(x)
---
exists x Smart(x)


--------------------------
exists x Student(x)
exists x Smart(x)
---
exists x (Student(x) ^ Smart(x))


--------------------------
--------------------------
Are the following pairs logically equivalent?
If not, show that they are not.

all x (P(x) ^ Q(x)),  
all x P(x) ^ all x Q(x)



--------------------------
all x (P(x) v Q(x)),
all x P(x) v all x Q(x).



--------------------------
exists x (P(x) v Q(x)),
exists x P(x) v exists x Q(x)

--------------------------
exists x (P(x) ^ Q(x)),
exists x P(x) ^ exists x Q(x)


Part III:=============================

Consider a world with only two shapes.  They are
both cubes and they are in the same row. Is the following
sentence true? [Note: this question was revised 3-17-09; it originally
said "both tets", a cut-and-paste mistake]

all x all y 
((cube(x) ^ cube(y)) --> (leftof(x,y) v rightof(x,y)))

----------------------------
Express the following sentences in English
(where taken(x,y) means that student x took class y):

exists x exists y taken(x,y)

exists x all y taken(x,y)

all x exists y taken(x,y)

exists y all x taken(x,y)

all y exists x taken(x,y)

exists y (small(y) ^  all x(small(y) --> y=x))

-------------

Suppose Q(x,y,z) is the statement x+y=z
The domain of discourse is the real numbers.

Are the following sentences true?

all x all y exists z Q(x,y,z)

exists z all x all y Q(x,y,z)

---------
Express the following sentences in logic
(where L(x,y) means that x loves y, and the domain of 
discourse for both x and y is the set of all people 
in the world)

everybody loves fred

every loves somebody

there is someone whom no one loves

everyone loves himself or herself

--------------------
Express the following in logic. First do this using
both function and predicate symbols, and then do it using only
predicate symbols.

Everyone has exactly one mother, who is a woman.