Exam Date: March 28, 2011
1. Show whether each of the following is a tautology, contradiction, or neither using truth tables:
(a) p → ¬( p ∨¬q)
(b) ¬p ∧ ¬q
(c) (( p ∨ ¬p) → q)
(d) ¬q ∨ (¬q → p)
(e) [( p ∨ q ) ∧ ( p ↔ r ) ∧ ( q ↔ r )]
2. Show whether each of the expression in problem 1 is a tautology, contradiction, or neither using the properties of logic. (logical equivalence's):
(a) 
Associative laws 
x(yz) = (xy)z 


x ∨ (y ∨ z) = (x ∨ y) ∨ z 
(b) 
Commutative laws 
x ∧ y = y ∧ x 


x ∨ y = y ∨ x 
(c) 
Distributive laws 
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) 


x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) 
(d) 
Identity laws 
x ∧ T = x 


x ∨ F = x 
(e) 
Dominance laws 
x ∧ F = F 


x ∨ T = T 
(f) 
Idempotent laws 
x ∧ x = x 


x ∨ x = x 
(g) 
Complement laws 
x ∧ ¬x = F 


x ∨ ¬x = T 
(h) 
Double complement law 
¬(¬x) = x 
(i) 
DeMorgan's laws 
¬(x ∧ y) = ¬x ∨ ¬y 


¬(x ∨ y) = ¬x ∧ ¬y 
3. Evaluate each of the following bitwise logic expressions
a. (011010 ∧ 111100) ∨ 001100 =
b. 111000 ∨ 001100 ∨ 000001 =
c. (101010 ∨ 110011) → 000111 =
4. Prove using truth tables whether each of the following pairs of expressions are equivalent:
a. p ↔ q and ( p ∨ q) → ( p ∧ q)
b. ( p ↔ q) ∨ p and ( p ∨ q) ∧ ¬( p ∧ q)
c. ¬( p ∧ ¬q) and p → q
5. Show whether each of the expression in problem 4 are equivalent using the properties of logic. (logical equivalence's):
6. State the converse and contrapositive of each of the following implications:
a. If it snows tonight, then I will stay home
b. I go to the beach whenever it is a sunny summer day.
c. When I stay up late, it necessary that I sleep until noon
Consider the following predicates:
C( x,y)  true if person x owns car y.
RED( x)  true if x is a red car.
BLUE( x)  true if x is a blue car.
7. Write the following in symbols:
a. Sandy owns a blue car.
b. Every car is either blue or red.
c. There is a car that is both blue and red.
d. Every blue car has an owner.
e. Every red car owner also owns a blue car
Now consider a world for the above predicates with only 4 people and four cars:
a. Bill owns a blue car and a red car.
b. Sandy owns a red car.
c. Robin owns a blue car.
d. Jack does not own a car.
For the following, consider ONLY the data above:
8. Tell whether each of the following is true or false, and why.
a. ∃ x : C(Bill, x)
b. ∃ x : C(Sandy, x) ∧ BLUE( x)
c. ∀x ∃y: C( x,y)
d. ∀y ∃x: C( x,y)
9. Suppose P(x,y) is the statement 2x < y, where x and y are integers. What are the truth values of
a) P(1, 1)
b) P(0, 0)
c) ∃y P(3,y)
d) ∀x∃y P( x,y)
e) ∃x∀ y P( x,y)
f) ∀y∃x P( x,y)
10. Suppose the variable x represents students and y represents dorm rooms:
R( x,y): student x is in dorm room y.
Write each of the following statements using the above predicate and any needed quantifiers:
a) Fran is in room Elmwood 321
b) Bill does not have a dorm room.
c) There is a dorm room which is not in use (no assigned students).
d) Every student has a dorm room.
e) There is a room with more then one student assigned to it.
11. Suppose P(x,y) is the statement x + 2y = xy, where x and y are integers. What are the truth values of
(a) P(1,1)
(b) P(0,0)
(c) ∃yP(3,y)
(d) ∀x∃yP( x,y)
(e) ∃x∀yP( x,y)
(f) ∀y∃xP( x,y)
(g) ∃y∀xP( x,y)
(h) ¬∀x∃y ¬P( x,y)
12. Find the size (that is, the cardinal number) of each of the following sets:
(a) {xx ∈ Z and x^{2} < 10}.
(b) P({a,b,c,d}), where P denotes the power set.
(c) {1,3,5,7...}.
(d) A x B where A = {1,2,3,4,5} and B = {1,2,3}.
(e) {xx ∈ N and 9x^{2}  1 = 0}.
(f) P(A), where A is the power set of {a,b,c}.
(g) A x B, where A = {a,b,c} and B = Ø.
(h) {xx ∈ N and 4x^{2}  1 = 0}.
13. Determine the truth value for each of the following.
(a) ∀x ∃y  (x^{2}=y)
(b) ∀x ∃y  (x=y^{2})
(c) ∃x ∀y  ( xy = 0)
(d) ∃x ∀y  (y ≠ 0 → xy = 1)
14. Suppose A = {a,b,c}. Mark each of the following TRUE or FALSE:
(a) {b,c} ⊆ P(A)
(b) {{a}} ⊆ P(A)
(c) Ø ⊆ A
(d) {Ø} ⊆ P(A)
(e) Ø ⊆ A x A
(f) {a,c} ∈ A
(g) {a,b} ∈ A x A
(h) (c,c) ∈ A x A
15.Find the size (that is, the cardinal number) of each of the following sets:
(a) {x x ∈ Z and x^{2} = 2x}
(b) P(A), where A = P(P({1,2,3})
(c) {1, 3, 5, 7...}
(d) S x T, where S = {a,b,c,d} and T = {1,2,3,4,5,6}
(e) {x  x ∈ Z and x^{2} < 80}.
(a) 1, (b) 2^(2^3)=256, (c) ¥ ,(d) 24, (e) 17
16. Let A = { 2,4,6,8 } B = {4, 7} C = {Ø, {4, 7}}
Show the following:
a. A ∪ B =
b. A ∩ B =
c. A ∪ C =
d. A ∩ C =
e. A  B =
f. C  Ø =
g. Ø  C =
h. C ∪ (A ∩ B) =
17. Show Venn Diagrams for each of the above.
18. Simplify the following using the properties of set operation:
19. Prove the following using a. set builder notation and b Membership tables
a. A ∩ (A ∪ B) = A
b. A ∩ (B ∪ C) = (A ∪ B) ∩ (A ∪ C)
 JimSkon  20110314