Midterm Review Sheet         Algebra               Winter 2005

 

About the Midterm: The midterm will start at 7 pm Wednesday, February 1 in our classroom.

You will have two hours to complete it. It will consist of some definitions to state, some theorems to

state, two theorems to prove, and one or two proof techniques to explain. In addition, there will

be some true/false questions for which you will be expected to briefly justify your answers. There

may be some brief examples to do.

Definitions (You should be able to give an example in each case)

 

  1.  Set operations: intersection, union, difference, Cartesian product
  2.  Complement of a set
  3.  Divides
  4.  Greatest common divisor and Least common multiple
  5.  Relatively prime
  6.  Well ordering principle
  7.  Operation on a set
  8.  Group
  9.  Abelian group
  10.  The group Zn
  11.  Order of a group
  12.  Direct product group
  13.  Subgroup
  14.  Subgroup generated by an element or a set
  15.  Cyclic subgroup of a group
  16.  Function, domain, range, image
  17.  Preimage of an element under a function
  18.  Injective function
  19.  Surjective function
  20.  Bijective function
  21.  Inverse function
  22.  Composition of functions
  23.  Permutation group of a set
  24. The group Sn
  25.  Dihedral group

 

Theorems to state

 

  1. Division algorithm
  2. Euclidean Algorithm

 

Theorems to prove

 

  1. Properties of divisibility  (Theorem 2 in appendix B)
  2. Inductive Principle
  3. If a set has an associative operation, then a string of products can be reassociated in any way.
  4. In a group, the identity element and inverses are unique.
  5. Zn is an abelian group.
  6. Cancellation law for groups.
  7. Theorem on inverses (Theorem 3, chapter 4)
  8. There is only one group table for a group of order 1, 2, or 3.
  9. Subgroup theorem (a subgroup of a group is itself a group).
  10. If a nonempty subset S of a finite group G is closed under the operation on G, then it is a subgroup.

 

 

Proof techniques to be able to explain

 

  1. Proof by Induction
  2. Proof by Strong Induction
  3. Proof by Contradiction
  4. Proof by Exhaustion (cases)
  5. Proof of Uniqueness
  6. Proof using uniqueness