أسواق
08-04-2013, 09:20 PM
M131: Discrete Mathematics
Q–1: [2+3 marks]
a) Determine whether each of the following is TRUE or FALSE:
i. If 27 3 (mod 4), then 4 | 32.
ii. Dogs can fly whenever 5 > 2.
iii. , domain is the set of real numbers.
iv. , domain is the set of integers.
b) Establish whether each of the following propositions is a tautology, contradiction or contingency:
i. (pq) ˅ (p q), by using truth table.
ii. (p (qp)), by using laws of logic.
Q–2: [2+3 marks]
a) Determine whether and are logically alent.
b) Let P(x) be the statement "x is happy" and Q(x) be the statement "x is under 20", where the universe of discourse for x is the set of AOU students. Express each of the following English statement in terms of quantifiers and logical connectives:
i. Some AOU students are unhappy.
ii. There is an AOU student who is happy and over 20.
iii. Not all AOU students are either unhappy or under 20.
Q¬–3: [3+2 marks]
a) Suppose the universal set U = {n: n is an integer and 1 ≤ n ≤ 11}, A = {x : x 6 (mod 2)}, B = {y : 3 | y}, and C = {3, 4, 5, 6, 7}.
i. List the elements of the sets A and B.
ii. List the elements of the set .
iii. How many members does the power set P(C) have? The Cartesian product A×B have?
b) By using set builder notation and logical alences, prove that
.
Q¬−4: [5×1 marks] Consider the decimal integers a = 51, b = 825 and c = 1638.
a) Find the prime factorizations of b and c.
b) Are a, b and c relatively prime? Explain.
c) Find gcd(a, b) and lcm(b, c).
d) Convert b to a binary number.
e) Convert c to a hexadecimal number.
Q−5: [2+3 marks]
a) Suppose that a computer has only the memory s 0, 1, 2, ..., 99. Use the hashing function h where h(x) = (x − 75) mod 100 to determine the memory s in which 7, 232, and 987 are stored.
b) Encrypt the message “GOOD ” using the encryption function
f (p) = (2p + 3) mod 26, 0 ≤ p ≤ 25.
Explain why f would not be a good coding function.
Q¬−6: [5×1 marks] Let R1 = {(x, y): x = y − 1} and R2 = {(x, y): x < y} be relations on the set A = {1, 2, 3, 4}.
a) List the elements of R1 and R2.
b) Find the matrix representation for each of the relations R1 and R2.
c) Find the symmetric closure of R1.
d) Find the transitive closure of R2.
e) Find .
Q¬−7: [5 marks] Let A be the set all ordered pairs of positive integers and R be the relation defined on A where (a, b)R(c, d) if and only if a + d = b + c. Show that R is an alence relation.
Q¬−8: [5×1 marks] Let R be the partial order relation defined on A = {2, 3, 6, 9, 10, 12, 14, 18, 20}, where xRy means x | y.
a) Draw the Hasse diagram for R.
b) Find all maximal and mammal elements.
c) Find lub({6,12}).
d) Find glb({6,12}).
e) What is the least element? The greatest element?
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
Q–1: [2+3 marks]
a) Determine whether each of the following is TRUE or FALSE:
i. If 27 3 (mod 4), then 4 | 32.
ii. Dogs can fly whenever 5 > 2.
iii. , domain is the set of real numbers.
iv. , domain is the set of integers.
b) Establish whether each of the following propositions is a tautology, contradiction or contingency:
i. (pq) ˅ (p q), by using truth table.
ii. (p (qp)), by using laws of logic.
Q–2: [2+3 marks]
a) Determine whether and are logically alent.
b) Let P(x) be the statement "x is happy" and Q(x) be the statement "x is under 20", where the universe of discourse for x is the set of AOU students. Express each of the following English statement in terms of quantifiers and logical connectives:
i. Some AOU students are unhappy.
ii. There is an AOU student who is happy and over 20.
iii. Not all AOU students are either unhappy or under 20.
Q¬–3: [3+2 marks]
a) Suppose the universal set U = {n: n is an integer and 1 ≤ n ≤ 11}, A = {x : x 6 (mod 2)}, B = {y : 3 | y}, and C = {3, 4, 5, 6, 7}.
i. List the elements of the sets A and B.
ii. List the elements of the set .
iii. How many members does the power set P(C) have? The Cartesian product A×B have?
b) By using set builder notation and logical alences, prove that
.
Q¬−4: [5×1 marks] Consider the decimal integers a = 51, b = 825 and c = 1638.
a) Find the prime factorizations of b and c.
b) Are a, b and c relatively prime? Explain.
c) Find gcd(a, b) and lcm(b, c).
d) Convert b to a binary number.
e) Convert c to a hexadecimal number.
Q−5: [2+3 marks]
a) Suppose that a computer has only the memory s 0, 1, 2, ..., 99. Use the hashing function h where h(x) = (x − 75) mod 100 to determine the memory s in which 7, 232, and 987 are stored.
b) Encrypt the message “GOOD ” using the encryption function
f (p) = (2p + 3) mod 26, 0 ≤ p ≤ 25.
Explain why f would not be a good coding function.
Q¬−6: [5×1 marks] Let R1 = {(x, y): x = y − 1} and R2 = {(x, y): x < y} be relations on the set A = {1, 2, 3, 4}.
a) List the elements of R1 and R2.
b) Find the matrix representation for each of the relations R1 and R2.
c) Find the symmetric closure of R1.
d) Find the transitive closure of R2.
e) Find .
Q¬−7: [5 marks] Let A be the set all ordered pairs of positive integers and R be the relation defined on A where (a, b)R(c, d) if and only if a + d = b + c. Show that R is an alence relation.
Q¬−8: [5×1 marks] Let R be the partial order relation defined on A = {2, 3, 6, 9, 10, 12, 14, 18, 20}, where xRy means x | y.
a) Draw the Hasse diagram for R.
b) Find all maximal and mammal elements.
c) Find lub({6,12}).
d) Find glb({6,12}).
e) What is the least element? The greatest element?
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645
حل واجب M131 منفرد [email protected] اتصل 0540893645 - 00966540893645