Math 360: Practice Test 1. NOTE: The test will differ from this practice test in the problems and their numbers, but
not in the types of problems
Definitions and Examples
I. Precisely define the following.
Group
Function
1-1 Function
Onto Function
Binary Operation
Math 360: Practice Test 1. NOTE: The test will differ from this practice test in the problems and their numbers, but
not in the types of problems
II. Examples and Counterexamples
1. Graphically and set-wise represent a relation with Domain D = {1,2,3} and Codomain {a,b,c,d} that is not a function
OR state why this is not possible.
2. Graphically and set-wise represent a function with Domain D = {1,2,3} and Codomain {a,b,c,d} that is 1-1 but not onto,
OR state why this is not possible.
3. Graphically and set-wise represent a function with Domain D = {1,2,3 } and Codomain {a,b,c} that is ONTO but not 11, OR state why this is not possible.
4. Using a table and set-wise represent a binary operation * on the set D = {a,b}, OR state why this is not possible.
Math 360: Practice Test 1. NOTE: The test will differ from this practice test in the problems and their numbers, but
not in the types of problems
III. Represent the group of symmetries of a rectangle two ways: As a set of matrices and as a set of permutations using
Cauchy notation. Draw lines to define an isomorphism between representations.
2
1
3
4
Matrix Representation
Permutation Representation
IV. Proofs and Problems
1. Prove or Disprove: Let (G,*) binary structure on G and let 0 and 1 be, respectively the additive and multiplicative
identity elements of G. If a*b = 0, then neither a nor b has an inverse in G.
2. Prove or disprove: Let (G,*) be a binary structure on G, and let e be the identity element in G. e is unique.
Math 360: Practice Test 1. NOTE: The test will differ from this practice test in the problems and their numbers, but
not in the types of problems
3. Using a table to represent the binary structures, demonstrate two contrasting examples of multiplication in Z modn:
one for which all elements have inverses; the other, where inverses are missing.
4. Suppose sets D and C are finite. State a counting argument that validates the following:
The number of possible functions from D to C is |C||D|.
5. There are only two types of abstract groups with only four elements: The Klein group and the Cyclic Group. In the
Klein Group, every element is its own inverse. In the Cyclic group, every element may be generated by a single
generator. (Here, think of either (Zmod 4,+) or the set of rotations of a square. Using S = (e,a,b,c}, write the correct g
binary structure table for each abstract group.
Klein Group
Cyclic Group