CS1502, Fall 2004

Some practice questions for Exam 2.  

This doesn't have questions covering all the things that will be on
the exam.  I think it will be useful practice to supplement the
lecture notes and assignments.

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, give a counter example showing it is not.

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, give an example that shows 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 tets and they are in the same row. Is the following
sentence true?

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.