discrete mathematics - exercises

M131 Problem Solving Session #1

Part 1: MULTIPLE CHOICE QUESTIONS

1. The converse of the proposition qp is

a) qp

b) pq

c) pq

d) qp e) None of the above

2. The negation of If is Friday, then it is cold is

a) If it is not Friday, then it is not cold b) It is not Friday or it is cold c) It is Friday and it is not cold d) If it is cold, then it is Friday e) None of the above

3. The negation of the statement All animals can fly i

a) All animals cannot fly. b) Some animals can fly. c) Some animals cannot fly. d) There exist an animal that can fly. e) None of the above.

4. If S = {1, 1, 1, 2, 3, 4, 4}, then |P(S)| =

a) 8 b) 16 c) 64 d) 256 e) None of the above

5. If C = {m, o, h, a, m, e, d}, then |P(C)| =


6. Which of the following is always true?

a) A B = A B, then A = B

b) A C = B C, then A = B

c) A C = B C, then A = B d) There is a set A such that P (A) = 12. e) None of the above


7. The converse of qp is a) qp

b) qp c) pq d) pq e) None of the above

8.

:

a) 0 b) 1 c) 2

d) e) None of the above

9. The inverse of the statement If you practice well, you will win the game is:

a) If you dont practice well, you will win the game. b) You will win the game if you practice well. c) If you dont win the game, then you did not practice well. d) If you win the game, then you must have practiced well. e) None of the above

10. Let S={a,b,c,d}, then |P(S)|:



Part 2: ESSAY QUESTIONS

1. [3+5 marks] a) Write the negation of the following statements:

i. It is neither cold nor dry. ii. If it is rainy, then we go to the movies.

b) Show that (P Q) R and P (Q R) are logically equivalent.

2. [2+3+3 marks] a) Consider the following predicates C(x): x is a check, T(x): x has been cashed within 30

days and V(x): x is void. Write every check is void if it has not been cashed for 30 days in terms of a quantifier and logical connectives.

b) Suppose U = {x: x is integer and 1 x 9}, A = {2, 4, 6, 8}, B = {3, 6, 9} and C = {3, 4, 5, 6, 7}. Find

i. C (B A).

ii. CBA . c) Show that ACBACAB )()()( .

3. [3+5 Marks] a) Determine whether each of the following is TRUE or False:

i. ii. 3 > 5 is sufficient for 1 + 1 = 2.

iii. xxx 2

, domain is the set of real numbers.

b) Using logic laws show that the proposition pqqp is a tautology. 4. [2+6 marks] a) Let I(x) and E(x) be the statements "x is an integer" and "x is even", respectively, domain

is the set of real numbers. Write Some integers are not even using quantifiers and logical connectives.

b) Let A = {a, b, c}. Determine whether each of the following is TRUE or FALSE:

iii. AAba },{ .

iv. APba },{ . v. APba },{ . vi. 8AP . vii. AA .

viii. 9 AAP .

5. a. [51 Marks] Determine whether each of the following is TRUE or FALE:

i. If (1 + 1 = 3), then (1 > 2 or 1 + 2 = 4). ii. (21 mod 4 = 2) and (4 | 16).

iii. xxx 11

, domain is the set of integers.


iv. xxx 23

, domain is the set of integers.

v. 111100111100 . b. [5 Marks] Using the truth table, determine whether the propositions

rqp and

qrp are logically equivalent.

6. (a) (2 marks) Show that the statement qpqp is a tautology (b) (3 marks) Write the converse, inverse, and contrapositive of the statement:

If 2x then 42 x

(c) (3 marks) Determine the truth value of each of the statements below:

(i) -2 > 0 whenever 3mod425

(ii) 0342 xxx where x is an integer. (iii) stp where p and s are true and t is false.

7. (a) (6 marks) Consider the universal set U= {x | x is an integer: 101 x } and the subsets

A, B, and C defined by:

A= number prime a is | xUx B= 2 ofpower a is | xUx C= {3,6,9}

(i) Write A and B in roster notation.

(ii) Find CBA , AB , and BAC . (iii) Find the set D with a bit string representation: 0011101110

8. (2 marks) Prove using De Morgans Laws that: BCBABCA 9.

a. (5 marks)

(i) Find the universal set U and the subsets A, and B if:

}5,1{ BA

}10,8,7,6,2{A

}10,9,3,2{B (Hint: Use a Venn Diagram)

(ii) Find the bit string representation of the set BA .

b. (3 marks) Re-write the following without any negations on quantifiers:


)( yxyx (ii) ))4()3(( yx

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