Negative points per question: Based on Section 4.2 of HMU. 1. h is a homomorphism from the alphabet {a,b,c} to {0,1}. If h(a) = 01, h(b) = 0, and h(c) = 10, which of the following strings is in h¯¹ (010010)? a) babc b) bacb c) cbbc d) abbc 2. The language of regular expression (0+10)* is the set of all strings of O's and 1's such that every 1 is immediately followed by a 0. Describe the complement of this language (with respect to the alphabet {0,1}) and identify in the list below the regular expression whose language is the complement of L((0+10)*). a) (0+1)*11(0+1)* + (0+10)*1 b) (0+10)*11(0+1)* c) (1+01)* d) (0+10)*1(ɛ+11(0+1)*) 3. If h is the homomorphism defined by h(a) = 0 and h(b) = ε, which of the following strings is in h¨¹ (000)? a) abaabaa b) abbbabaab c) abaabbb d) babab 4. Find, in the list below, a regular expression whose language is the reversal of the language of this regular expression: 10(2+3). Recall that the reversal of a language is formed by reversing all its strings, and the reversal of a string a₁a2…..a is aɲ…..a2a1. a) 01(2+3) b) 0(2+3)1 c) (2+3)01 d) 1(2+3)0 5. Let h be the homomorphism defined by h(a) = 01, h(b) = 10, h(c) = 0, and h(d) = 1. If we take any string w in (0+1)*, h¯¹(w) contains some number of strings, N(w). For example, h¨¹ (1100) = {ddcc, dbc}, i.e., N(1100) = 2. We can calculate the number of strings in h¨¹ (w) by a recursion on the length of w. For example, if w= 00x for some string x, then N(w) = N(0x), since the first 0 in w can only be produced from c, not from a. Complete the reasoning necessary to compute N(w) for any string w in (0+1)*. Then, choose the correct value of N(110011001). a) 55 b) 16 c) 8 d) 256 6. The operation DM(L) is defined as follows: 1. Throw away every even-length string from L. 2. For each odd-length string, remove the middle character. For example, if L = {001, 1100, 10101}, then DM(L) = {01, 1001}. That is, even-length string 1100 is deleted, the middle character of 001 is removed to make 01, and the middle character of 10101 is removed to make 1001. It turns out that if L is a regular language, DM(L) may or may not be regular. For each of the following languages L, determine what DM(L) is, and tell whether or not it is regular. • L₁: the language of regular expression (01)*0. • L2: the language of regular expression (0+1)*1(0+1)*. • L3: the language of regular expression (101)*. • L4: the language of regular expression 00*11*. Now, identify the true statement below. a) DM(L1) is not regular; it consists of all strings of the form (01) (00+ε) (10)". b) DM(L2) is regular; it is the language of regular expression (0+1)*. c) DM(L2) is not regular; it consists of all even-length strings with more O's than 1's. d) DM(L1) is not regular; it consists of all strings of the form (01) (00) (10)".

Negative points per question: Based on Section 4.2 of HMU. 1. h is a homomorphism from the alphabet {a,b,c} to {0,1}. If h(a) = 01, h(b) = 0, and h(c) = 10, which of the following strings is in h¯¹ (010010)? a) babc b) bacb c) cbbc d) abbc 2. The language of regular expression (0+10)* is the set of all strings of O's and 1's such that every 1 is immediately followed by a 0. Describe the complement of this language (with respect to the alphabet {0,1}) and identify in the list below the regular expression whose language is the complement of L((0+10)*). a) (0+1)*11(0+1)* + (0+10)*1 b) (0+10)*11(0+1)* c) (1+01)* d) (0+10)*1(ɛ+11(0+1)*) 3. If h is the homomorphism defined by h(a) = 0 and h(b) = ε, which of the following strings is in h¨¹ (000)? a) abaabaa b) abbbabaab c) abaabbb d) babab 4. Find, in the list below, a regular expression whose language is the reversal of the language of this regular expression: 10(2+3). Recall that the reversal of a language is formed by reversing all its strings, and the reversal of a string a₁a2…..a is aɲ…..a2a1. a) 01(2+3) b) 0(2+3)1 c) (2+3)01 d) 1(2+3)0 5. Let h be the homomorphism defined by h(a) = 01, h(b) = 10, h(c) = 0, and h(d) = 1. If we take any string w in (0+1)*, h¯¹(w) contains some number of strings, N(w). For example, h¨¹ (1100) = {ddcc, dbc}, i.e., N(1100) = 2. We can calculate the number of strings in h¨¹ (w) by a recursion on the length of w. For example, if w= 00x for some string x, then N(w) = N(0x), since the first 0 in w can only be produced from c, not from a. Complete the reasoning necessary to compute N(w) for any string w in (0+1)*. Then, choose the correct value of N(110011001). a) 55 b) 16 c) 8 d) 256 6. The operation DM(L) is defined as follows: 1. Throw away every even-length string from L. 2. For each odd-length string, remove the middle character. For example, if L = {001, 1100, 10101}, then DM(L) = {01, 1001}. That is, even-length string 1100 is deleted, the middle character of 001 is removed to make 01, and the middle character of 10101 is removed to make 1001. It turns out that if L is a regular language, DM(L) may or may not be regular. For each of the following languages L, determine what DM(L) is, and tell whether or not it is regular. • L₁: the language of regular expression (01)*0. • L2: the language of regular expression (0+1)*1(0+1)*. • L3: the language of regular expression (101)*. • L4: the language of regular expression 00*11*. Now, identify the true statement below. a) DM(L1) is not regular; it consists of all strings of the form (01) (00+ε) (10)". b) DM(L2) is regular; it is the language of regular expression (0+1)*. c) DM(L2) is not regular; it consists of all even-length strings with more O's than 1's. d) DM(L1) is not regular; it consists of all strings of the form (01) (00) (10)".

Negative points per question: Based on Section 4.2 of HMU. 1. h is a homomorphism from the alphabet {a,b,c} to {0,1}. If h(a) = 01, h(b) = 0, and h(c) = 10, which of the following strings is in h¯¹ (010010)? a) babc b) bacb c) cbbc d) abbc 2. The language of regular expression (0+10)* is the set of all strings of O's and 1's such that every 1 is immediately followed by a 0. Describe the complement of this language (with respect to the alphabet {0,1}) and identify in the list below the regular expression whose language is the complement of L((0+10)). a) (0+1)11(0+1)* + (0+10)1 b) (0+10)11(0+1)* c) (1+01)* d) (0+10)1(ɛ+11(0+1)) 3. If h is the homomorphism defined by h(a) = 0 and h(b) = ε, which of the following strings is in h¨¹ (000)? a) abaabaa b) abbbabaab c) abaabbb d) babab 4. Find, in the list below, a regular expression whose language is the reversal of the language of this regular expression: 10(2+3). Recall that the reversal of a language is formed by reversing all its strings, and the reversal of a string a₁a2…..a is aɲ…..a2a1. a) 01(2+3) b) 0(2+3)1 c) (2+3)01 d) 1(2+3)0 5. Let h be the homomorphism defined by h(a) = 01, h(b) = 10, h(c) = 0, and h(d) = 1. If we take any string w in (0+1), h¯¹(w) contains some number of strings, N(w). For example, h¨¹ (1100) = {ddcc, dbc}, i.e., N(1100) = 2. We can calculate the number of strings in h¨¹ (w) by a recursion on the length of w. For example, if w= 00x for some string x, then N(w) = N(0x), since the first 0 in w can only be produced from c, not from a. Complete the reasoning necessary to compute N(w) for any string w in (0+1). Then, choose the correct value of N(110011001). a) 55 b) 16 c) 8 d) 256 6. The operation DM(L) is defined as follows: 1. Throw away every even-length string from L. 2. For each odd-length string, remove the middle character. For example, if L = {001, 1100, 10101}, then DM(L) = {01, 1001}. That is, even-length string 1100 is deleted, the middle character of 001 is removed to make 01, and the middle character of 10101 is removed to make 1001. It turns out that if L is a regular language, DM(L) may or may not be regular. For each of the following languages L, determine what DM(L) is, and tell whether or not it is regular. • L₁: the language of regular expression (01)0. • L2: the language of regular expression (0+1)1(0+1). • L3: the language of regular expression (101). • L4: the language of regular expression 0011. Now, identify the true statement below. a) DM(L1) is not regular; it consists of all strings of the form (01) (00+ε) (10)". b) DM(L2) is regular; it is the language of regular expression (0+1)*. c) DM(L2) is not regular; it consists of all even-length strings with more O's than 1's. d) DM(L1) is not regular; it consists of all strings of the form (01) (00) (10)".

Elements Of Modern Algebra

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.4: Binary Operations

Problem 3E

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Related questions

Question

Negative points per question:
Based on Section 4.2 of HMU.
1. h is a homomorphism from the alphabet {a,b,c} to {0,1}. If h(a) = 01, h(b) = 0,
and h(c) = 10, which of the following strings is in h¯¹ (010010)?
a) babc
b) bacb
c) cbbc
d) abbc
2. The language of regular expression (0+10)* is the set of all strings of O's and
1's such that every 1 is immediately followed by a 0. Describe the complement
of this language (with respect to the alphabet {0,1}) and identify in the list
below the regular expression whose language is the complement of
L((0+10)*).
a) (0+1)*11(0+1)* + (0+10)*1
b) (0+10)*11(0+1)*
c) (1+01)*
d) (0+10)*1(ɛ+11(0+1)*)
3. If h is the homomorphism defined by h(a) = 0 and h(b) = ε, which of the
following strings is in h¨¹ (000)?
a) abaabaa
b) abbbabaab
c) abaabbb
d) babab
4. Find, in the list below, a regular expression whose language is the reversal of
the language of this regular expression: 10(2+3). Recall that the reversal of a
language is formed by reversing all its strings, and the reversal of a string
a₁a2…..a is aɲ…..a2a1.
a) 01(2+3)
b) 0(2+3)1
c) (2+3)01
d) 1(2+3)0

5. Let h be the homomorphism defined by h(a) = 01, h(b) = 10, h(c) = 0, and h(d)
= 1. If we take any string w in (0+1)*, h¯¹(w) contains some number of strings,
N(w). For example, h¨¹ (1100) = {ddcc, dbc}, i.e., N(1100) = 2. We can
calculate the number of strings in h¨¹ (w) by a recursion on the length of w. For
example, if w= 00x for some string x, then N(w) = N(0x), since the first 0 in w
can only be produced from c, not from a.
Complete the reasoning necessary to compute N(w) for any string w in (0+1)*.
Then, choose the correct value of N(110011001).
a) 55
b) 16
c) 8
d) 256
6. The operation DM(L) is defined as follows:
1. Throw away every even-length string from L.
2. For each odd-length string, remove the middle character.
For example, if L = {001, 1100, 10101}, then DM(L) = {01, 1001}. That is,
even-length string 1100 is deleted, the middle character of 001 is removed to
make 01, and the middle character of 10101 is removed to make 1001.
It turns out that if L is a regular language, DM(L) may or may not be regular.
For each of the following languages L, determine what DM(L) is, and tell
whether or not it is regular.
• L₁: the language of regular expression (01)*0.
•
L2: the language of regular expression (0+1)*1(0+1)*.
• L3: the language of regular expression (101)*.
• L4: the language of regular expression 00*11*.
Now, identify the true statement below.
a) DM(L1) is not regular; it consists of all strings of the form (01) (00+ε)
(10)".
b) DM(L2) is regular; it is the language of regular expression (0+1)*.
c) DM(L2) is not regular; it consists of all even-length strings with more O's
than 1's.
d) DM(L1) is not regular; it consists of all strings of the form (01) (00)
(10)".

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Assume that is a binary operation on a non empty set A, and suppose that is both commutative and associative. Use the definitions of the commutative and associative properties to show that [ (ab)c ]d=(dc)(ab) for all a,b,c and d in A.
Suppose that the check digit is computed as described in Example . Prove that transposition errors of adjacent digits will not be detected unless one of the digits is the check digit. Example Using Check Digits Many companies use check digits for security purposes or for error detection. For example, an the digit may be appended to a -bit identification number to obtain the -digit invoice number of the form where the th bit, , is the check digit, computed as . If congruence modulo is used, then the check digit for an identification number . Thus the complete correct invoice number would appear as . If the invoice number were used instead and checked, an error would be detected, since .
In the -letter alphabet A described in Example, use the translation cipher with key to encipher the following message. the check is in the mail What is the inverse mapping that will decipher the ciphertext? Example 2 Translation Cipher Associate the letters of the "alphabet" with the integers. Let and define the mapping by where is the key, the number of positions from the plaintext to the ciphertext. If our alphabet consists of through, in natural order, followed by a blank, then we have "letters" that we associate with the integers as follows:

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