Math 3321                                Review Test #1                 Fall 2005

Dr. Cordero

1. Give the following definitions:
• Mapping (include the definition of domain, codomain, image)
• Onto mapping
• One-to-one mapping
• Composition of mappings
• Inverse of a mapping
• Invertible mapping

1. State and prove Theorem 2.1.
2. State and prove Theorem 2.2.
3. Complete the following definitions:

• An operation  on a set S is .....
• A set S is closed with respect to an operation  if ….
• An operation on a set S is associative if…
• An operation  on a set S is commutative if …
• An element  is an identity for an operation  on S if …
• An element is an inverse for relative to an operation  on S if ….

1. State and prove Theorem 4.1.
2. Give the following definitions:
• Permutation
• Permutation group
• Abelian group

1. State and prove Theorem 6.1.
2. State and prove Theorem 6.2.
3. State and prove Theorem 6.3.
4. Give the definition of subgroup.
5. State and prove Lemma 7.1.
6. State and prove Theorem 7.1.
7. State as a theorem Problem 7.22. Prove your theorem.