Assignment (100 marks)

1) Which of the following sentences are propositions? What are the truth values of those that

are propositions? (5 marks)

(a) Open the door.

(b) I like mathematics.

(c) 2 + 4 = 6

(d) X + 3 = 12

(e) Will you pass the examination?

2) What is the negation of each of these propositions? (3 marks)

(a) Charles has a smartphone.

(b) There is no pollution in London.

(c) 3+2= 5

3) State the converse, contrapositive and inverse of each of these conditional statements:

(4 marks)

(a) If it snows today, I will ski tomorrow.

(b) I come to class whenever there is going to be a test.

4) Construct a truth table for each of these compound propositions: (6 marks)

(a) (p q) (q p)

(b) (p q) (p q)

(c) (p q) (q r)

5) Translate the given statements into propositional logic using the propositions provided:

(a) You cannot edit a protected site entry unless you are an administrator. (1 mark)

Let p be the proposition statement You can edit a protected site,

and q be the proposition statement You are the administrator.2

5) (cont.)

(b) You can graduate if you have completed the requirements of your course and you do

not owe money to the university and you do not have an overdue library loan.

(2 marks)

Let p be the proposition statement You can graduate,

q be the proposition statement You owe money to the university,

r be the proposition statement You have completed the requirements of your course,

and s be the proposition statement You have an overdue library book.

6) (a) Show that p q and (p q) (p q) are logically equivalent. (3 marks)

(b) Show that p (q r) and q (p r) are logically equivalent. (4 marks)

7) Show that each of these conditional statements is a tautology by using truth tables:

(a) (pq)q (2 marks)

(b) (pq) q (2 marks)

8) What is the cardinality of each of these sets: (4 marks)

(a) {x}

(b) {{x}}

(c) {x,{x}}

(d) {x,{x},{x,{x}}}

9) Find the power set of each of these sets, where x, y are distinct elements: (4 marks)

(a) {x, y}

(b) {x, y, z}

10) Let A={a,b,c,d} and B={x,y}. Find: (4 marks)

(a) A x B

(b) B x A

11) Let A = {1,2,3,4,5} and B={0,3,6}. Find: (4 marks)

(a) AB

(b) AB

(c) A-B

(d) B-A

12) Draw the Venn diagrams for each of these combinations of the sets A, B, and C:

(a) A(B-C) (2 marks)

(b) (AB)(AC) (3 marks)3

13) Let f (x) = x2 + 1 and g(x) = x + 2 are relations on the set of real numbers.

(a) Decide whether f(x) and g(x) are function, if so, state whether they are injection,

surjection or bijection. (4 marks)

(b) Does either f (x) or g(x) have an inverse? If so, find the inverse. (3 marks)

(c) Find f g and g f . (6 marks)

14) For each of the following relations on the set {1, 2, 3, 4}, decide whether it is reflexive,

whether it is symmetric, whether it is antisymmetric, and whether it is transitive:

(6 marks)

a) {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)}

b) {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}

c) {(2, 4), (4, 2)}

15) Let A = {7,9,10,12}, B = {1,2,3,4} and consider the relations:

R = {(a,b)AB | a-2b 3}

G = {(a,b) | aA , bB and (a+b) = 12}

H = {(a,b) | aB , bB and (a+b) > 5}

(a) List all the elements of the relation R and give its cardinality |R|. (4 marks)

(b) Find the range and domain of the relation R. (2 marks)

(c) Find the inverse relation R-1. (2 marks)

(d) Find the sets of ordered pair in G and H. (6 marks)

(e) Draw the diagraphs of G and H. (6 marks)

(f) Give the Boolean-matrix representations of G and H. (2 marks)

16) Determine whether each of these functions is a bijection from to , where is the set

of real numbers, justify your answers: (6 marks)

(a) f(x)=2x+1

(b) f(x)=x2 +1

(c) f(x)=x3

GS407 Mathematics for Computer Science Assignment (100 marks) 5) (cont.) (6) You can graduate if you have completed the requirements of your course and you do not owe money to the university and you do not have an overdue library loan. . (2 marks) 1.et phe the proposition statement "You can graduate", he the proposition statement "You owe money to the university", r be the proposition statement "You have completed the requirements of your course", and s be the proposition statement "You have an overdue library book" 1) Which of the following sentences are propositions? What are the truth values of those that are propositions? (5 marks) 6) (a) Show that p ++4 and (pA) (PA) are logically equivalent. (6) Show that p + (q 1) and q + p VT) ure logicully equivalent. (marks) (4 marks) (a) Open the door. (b) I like mathematics. (c) 2 + 4-6 (d) x + 3 = 12 (e) Will you pass the exumination 7) Show that each of these conditional statements is a tautology by using truth tables: (a) (pag- (2 marks) (b) p4)+-4 (2 marks) 2) What is the negation of each of these propositions? (3 marks) (4 murks) ) (a) Charles has a smartphone. (b) There is no pollution in London. (c) 3+25 8) What is the cardinality of each of these sets: (a) x (b) {x}} (c) {x,{x}} (d) {x{x},{x,{x}}) 3) State the converse, contrapositive and inverse of each of these conditional statements: (4 marks) (a) If it snows today, I will ski tomorrow. (b) I come to class whenever there is yoing to be a test. (4 murks) 9) Find the power sel of each of these sels, where x, y are distinct elements: (a) (x,y} (b) x, y, z) 4) Construct a truth table for each of these compound propositions: (6 marks) (4 marks) (a) (p 9) (9 p) (b) (pv) + (p) (c) (P+)(4+) 10) Let A-{a,b,c,d) and B-{x,y). Find: (a) AB (b) BXA 5) Translate the given statements into propositional logic using the propositions provided: (4 murks) (a) You cannot edit a protected site entry unless you are an administrator. (1 mark) 11) Let A = {1,2,3,4,5) and B=0,3,6). L'ind: (a) AUB (b) AB (c) A-B (d) B-A Lep be the proposition statement "You can edit a protected site", and 4 he the proposition latement "You are the administralur