BELet S = {0,1,2). Define * such that a*b = |a - b for a, b E S.Which statement best describes problem?It is a binary operation with an identity.It is not a binary operation.It is an associative binary operation.It is a binary operation with two idempotents.It is a binary operation with a zero.

Answered: Image /qna-images/answer/33b60c53-4de3-4728-b6b4-91a1b66083d8.jpg

Answered: B E Let S = {0,1,2). Define * such that…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.4: Binary Operations

Problem 4TFE: True or False Label each of the following statements as either true or false. 4. Let . The empty set...

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Assume that is an associative binary operation on A with an identity element. Prove that the inverse of an element is unique when it exists.
In each part following, a rule that determines a binary operation on the set of all integers is given. Determine in each case whether the operation is commutative or associative and whether there is an identity element. Also find the inverse of each invertible element. b. d. f. h. j. l. for n. for
9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of all even integers is closed with respect to a. addition defined on . b. multiplication defined on .
4. Let be the relation “congruence modulo 5” defined on as follows: is congruent to modulo if and only if is a multiple of , and we write . a. Prove that “congruence modulo ” is an equivalence relation. b. List five members of each of the equivalence classes and .
5. Let be the relation “congruence modulo ” defined on as follows: is congruent to modulo if and only if is a multiple of , we write . a. Prove that “congruence modulo ” is an equivalence relation. b. List five members of each of the equivalence classes and .
True or False Label each of the following statements as either true or false. 4. Let . The empty set is the identity element in with respect to the binary operation .
10. Prove or disprove that the set of all nonzero integers is closed with respect to a. addition defined on . b. multiplication defined on .
Prove that addition is associative in Q.
True or false Label each of the following statement as either true or false. The greatest common divisor is as binary operation from to .
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 .
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.

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