Elementary Discrete Math

Sample Exam I

Exam Date: March 28, 2011

1. Show whether each of the following is a tautology, contradiction, or neither using truth tables:

(a) p → ( pq)

(b) pq

(c) (( pp) → q)

(d) q ∨ (qp)

(e) [( p q ) ∧ ( pr ) ∧ ( qr )]


2. Show whether each of the expression in problem 1 is a tautology, contradiction, or neither using the properties of logic. (logical equivalence's):

Logical Equivalences Table

(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. pq and ( pq) → ( pq)

b. ( pq) ∨ p and ( pq) ∧ ( pq)

c. ( pq) and pq


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) ∀xy P( x,y)

e) ∃xy P( x,y)

f) ∀yx 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) ∃xyP( x,y)

(f) ∀yxP( x,y)

(g) ∃yxP( x,y)

(h) ∀xy P( x,y)


12. Find the size (that is, the cardinal number) of each of the following sets:

(a) {x|x ∈ Z and x2 < 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) {x|x ∈ N and 9x2 - 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 = .

(h) {x|x ∈ N and 4x2 - 1 = 0}.


13. Determine the truth value for each of the following.

(a) ∀x ∃y | (x2=y)


(b) ∀xy | (x=y2)



(c) ∃xy | ( xy = 0)

(d) ∃xy | (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) ⊆ A

(d) {} ⊆ P(A)

(e) ⊆ 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 x2 = 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 x2 < 80}.

(a) 1, (b) 2^(2^3)=256, (c) ,(d) 24, (e) 17


16. Let A = { 2,4,6,8 } B = {4, 7} C = {, {4, 7}}

Show the following:

a. A ∪ B =

b. A ∩ B =

c. A ∪ C =

d. A ∩ C =

e. A - B =

f. C - =

g. - C =

h. C ∪ (A ∩ B) =


17. Show Venn Diagrams for each of the above.


18. Simplify the following using the properties of set operation:

  1. (A ∪ B)c ∩. (A c ∩ B c)
  2. (A ∪ (B ∩ A))
  3. (A ∪ (B ∩ A)) c ∩ ((C c ∪ B c) ∩ A c) c

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 - 2011-03-14

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Topic revision: r2 - 2011-03-14 - JimSkon
 
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