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![Prove that if * is any commutative and associative binary operation on a set S, then (a * b) * (c * d) =
[(d * c) * a] * b.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78da6f0a-16ad-4360-bd0d-c2290a710465%2F7ea96874-e618-4660-85ca-cb1f28d7f40f%2Fyme20s7_processed.jpeg&w=3840&q=75)
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- 15. Let be a binary operation on the non empty set . Prove that if contains an identity element with respect to , the identity element is unique.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.True or False Label each of the following statements as either true or false. 1. If a binary operation on a nonempty set is commutative, then an identity element will exist in .
- True or False Label each of the following statements as either true or false. 2. If * is a binary operation on a nonempty set , then is closed with respect to *.True or False Label each of the following statements as either true or false. 7. Any binary operation defined on a set containing a single element is commutative and associative.In Exercises , prove each statement. 17. if and only if .
- 43. Let the operation of addition be as defined in Exercise 42. Prove each of the following statements. a. b. 42. Let the operation of addition be defined on subsets and of by9. 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 .