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Mappings 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) Let and be mappings from to defined by and for each Which of these mappings are onto? Which of these mappings are one-to-one? Determine the range of   and

(2) How many mappings are there from onto How many one-to-one mappings are there from to

(3) How many one-to-one mappings are there from to

(4) For each is defined by For which values of is onto? For which values of is one-to-one?

(5) Let be a mapping and let Prove that is onto if and only if

(6) Prove that if and and are subsets of then

(7) Prove that if and and are subsets of then Then show that equality need not hold.

(8) Prove that a mapping is one-to-one if and only if for every pair of subsets and of

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