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Operations Homework

Directions: Write your homework legibly with a pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution/proof and show all steps. Do not turn in your scratch work. Staple your pages together, with the problems in the correct order, and use this page as a cover sheet with your name at the top right.

(1) Does define an operation on the set of all integers?

(2) Complete the following table in a way that makes commutative.

(3) How many different commutative operations are there on a one-element set? A two-element set? A three-element set? An -element set?

(4) Complete the following table in a way such that becomes an identity element. In how many ways can this be done?



Can the table be completed in a way such that and both become identity elements? Why or why not? Prove that an operation on a set (any ) can have at most one identity element?

(5) Complete the following table in such a way that is commutative and has an identity element, and each element has an inverse. How many ways can this be done?

(6) Assume that is an operation on with identity element Prove that to check whether is associative on it suffices to check the associativity condition for the non-identity elements of that is, is automatically true if any one of or is equal to

(7) Prove: If is an associative operation on then









for any and

(8) Assume that is an operation on with identity element and that for all Prove that is commutative and associative.

(9) Let Define the operation on by Find the identity element of with respect to Prove that is commutative and associative. For each element find its inverse.

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