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Composition as an Operation 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) With the set contains four elements, denote these by and defined as follows



Construct the Cayley table for composition as an operation on Which is the identity element? Is commutative as an operation on Which elements of are invertible? Is commutative as an operation on the set of invertible elements in

(2) With and let where and are elements of defined  as follows



Construct a Cayley table for A with respect to Is closed, associative, and commutative? Which is the identity element?

(3) Let denote the set of all real numbers except and Consider the six mappings form to defined as follows



Verify that composition is an operation on by constructing a Cayley table. Determine the identity element. Show that each element has an inverse. Show that is associative and commutative.  

(4) Prove that if and are any mappings, then Write a statement about what this means.

(5) Prove that if and are any mappings with and , then Write a statement about what this means.

(6) For in define by Show that function composition on the set is an operation which is associative. For each element find its inverse. Find the identity. Is this commutative?

(7) For a fixed point in define by Show that function composition on the set is an operation which is associative. For each element find its inverse. Find the identity. Is this commutative?

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