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“ef”), (“cd”, “pq”)}. Let...

1. Let R be the relation {(“ cd”, “ef”), (“ab”, “cd”), (“ab”, “ef”), (“cd”, “pq”)}. Let A be the set {“cd”, “rs”} .

Which statement below is correct:

a) R(A) = (“ef”, “pq”)

b) R(A) = {“ef”, “pq”}

c) R(A) = {(“cd”, “ef”), (“cd”, “pq”)}

d) R(A) = {“ab”}

2. Let R be the relation {(“ cd”, “ef”), (“ab”, “cd”), (“ab”, “ef”), (“cd”, “pq”)}. Let A be the set {“cd”, “rs”} .

Which statement below is correct:

a) A|R|A = R*

b) A|R = {(“ab”, “cd”)}

c) A|R|A = { }

d) R|A = {(“cd”, “ef”), (“cd”, “pq”)}

3. Suppose ELEM(n, s) refers to the nth character of the string s (assuming that LEN(s), the length of the string s, is at least n), and DELETE (n, s) is a function that outputs a string with the nth character deleted. Suppose further that we say a character αis greater that a character β, written as α> β, if the ASCII value of αis greater than that of β. An algorithm ALG F to evaluate a function F, which takes in a non-empty string s and output a character m, is given below.

Algorithm F

if LEN(s) = 1 then m := ELEM(1, s) (1)

else

temp := F(DELETE(2, s)) (2)

if ELEM(1, s) temp then m := temp (4)

else m := temp (5)

Let T(n) denotes the number of comparisons (i.e. =, >, β, if the ASCII value of αis greater than that of β. An algorithm ALG F to evaluate a function F, which takes in a non-empty string s and output a character m, is given below.

Algorithm F

if LEN(s) = 1 then m := ELEM(1, s) (1)

else

temp := F(DELETE(2, s)) (2)

if ELEM(1, s) temp then m := temp (4)

else m := temp (5)

Which statement below is correct:

a) F(“abc”) = ‘a’

b) F(“abc”) = “a”

c) F(“abc”) = ‘c’

d) F(“abc”) = ‘b

5. Let R be the relation {(“ cd”, “ef”), (“ab”, “cd”), (“ab”, “ef”), (“cd”, “pq”)}. Let A be the set {“cd”, “rs”} .

Which statement below is correct:

Answer

a) R* = {(“ab”, “cd”), (“cd”, “ef”), (“cd”, “pq”), (“ab”, “ef”), (“ab”, “pq”)}

b) R = R*

c) R is transitive

d) R2 = { }

6. The time complexity functions, f, g and h, of three different algorithms A, B and C to evaluate the same function are given by

for algorithm A, f(n) = - n3 +(NNN) NNN-NNNN2

for algorithm B, g(n) = 10-100 n3 + 80n

for algorithm C, h(n) = en

By classifying each of the functions f, g and h as O(F(n)) for a suitable function F, which statement below is correct:

Answer

a) The three algorithms are equally efficient

b) Algorithm B is less efficient than Algorithm C

c) Algorithm A and B are, in the rough sense, equally efficient when n is large

d) Algorithm A is less efficient than Algorithm C

Which statement below is correct:

a) R(A) = (“ef”, “pq”)

b) R(A) = {“ef”, “pq”}

c) R(A) = {(“cd”, “ef”), (“cd”, “pq”)}

d) R(A) = {“ab”}

2. Let R be the relation {(“ cd”, “ef”), (“ab”, “cd”), (“ab”, “ef”), (“cd”, “pq”)}. Let A be the set {“cd”, “rs”} .

Which statement below is correct:

a) A|R|A = R*

b) A|R = {(“ab”, “cd”)}

c) A|R|A = { }

d) R|A = {(“cd”, “ef”), (“cd”, “pq”)}

3. Suppose ELEM(n, s) refers to the nth character of the string s (assuming that LEN(s), the length of the string s, is at least n), and DELETE (n, s) is a function that outputs a string with the nth character deleted. Suppose further that we say a character αis greater that a character β, written as α> β, if the ASCII value of αis greater than that of β. An algorithm ALG F to evaluate a function F, which takes in a non-empty string s and output a character m, is given below.

Algorithm F

if LEN(s) = 1 then m := ELEM(1, s) (1)

else

temp := F(DELETE(2, s)) (2)

if ELEM(1, s) temp then m := temp (4)

else m := temp (5)

Let T(n) denotes the number of comparisons (i.e. =, >, β, if the ASCII value of αis greater than that of β. An algorithm ALG F to evaluate a function F, which takes in a non-empty string s and output a character m, is given below.

Algorithm F

if LEN(s) = 1 then m := ELEM(1, s) (1)

else

temp := F(DELETE(2, s)) (2)

if ELEM(1, s) temp then m := temp (4)

else m := temp (5)

Which statement below is correct:

a) F(“abc”) = ‘a’

b) F(“abc”) = “a”

c) F(“abc”) = ‘c’

d) F(“abc”) = ‘b

5. Let R be the relation {(“ cd”, “ef”), (“ab”, “cd”), (“ab”, “ef”), (“cd”, “pq”)}. Let A be the set {“cd”, “rs”} .

Which statement below is correct:

Answer

a) R* = {(“ab”, “cd”), (“cd”, “ef”), (“cd”, “pq”), (“ab”, “ef”), (“ab”, “pq”)}

b) R = R*

c) R is transitive

d) R2 = { }

6. The time complexity functions, f, g and h, of three different algorithms A, B and C to evaluate the same function are given by

for algorithm A, f(n) = - n3 +(NNN) NNN-NNNN2

for algorithm B, g(n) = 10-100 n3 + 80n

for algorithm C, h(n) = en

By classifying each of the functions f, g and h as O(F(n)) for a suitable function F, which statement below is correct:

Answer

a) The three algorithms are equally efficient

b) Algorithm B is less efficient than Algorithm C

c) Algorithm A and B are, in the rough sense, equally efficient when n is large

d) Algorithm A is less efficient than Algorithm C

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