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Abstract Algebra One (Exam One)

(1) Define (a) the Cartesian product of two sets, (b) a mapping between two sets, and (c) a binary operation on a set.

Cartesian Product:

Mapping:

Binary Operation:

(2) Using a Cayley table, give an example of a commutative group of order 5. Explain why it is a group.

(3) Give an example of a mapping that is one-to-one and not onto. Justify your claims.

(4) Give an example of a mapping that is onto and not one-to-one. Justify your claims.

(5) If where and are subsets of then show is one-to-one if and only if

(6) Prove there is at most one identity element under a binary operation on a set

(7) Prove that every invertible mapping has a unique inverse.

(8) Let be a non-empty subset of Prove that is a subgroup of a group if and only if for all

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