Math 360, Fall 2014, Assignment 3
By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so, worthy of my consideration.
- - Mary Shelley, Frankenstein
Carefully define the following terms, then give one example and one non-example of each:
- Isomorphism (from one binary structure to another).
- Isomorphic (binary structures).
- Structural property (of a binary structure).
- Identity element.
- Inverse (of an element of some binary structure with an identity).
- Group.
- Abelian group.
Carefully state the following theorems (you do not need to prove them):
- Theorem concerning the uniqueness of identity elements (Theorem 3.13).
- Theorem concerning the uniqueness of inverses in groups (second part of Theorem 4.17).
Solve the following problems:
- Section 3, problems 2, 4, 8, 9, and 17.
- Section 4, problems 1, 5, 7, 12, 13, and 17.
Questions:
Solutions:
In an effort to promote collaboration, I will only post work on problems whose problem numbers are elements of [2]3. Please join me in working on the problems here. -Ian
Definitions:
- A structural property of a binary structure (S,∗) is a property that holds for all structures that are isomorphic to (S,∗).
- A group (G,∗) is a binary structure for which the following three properties hold:
- ∗ is associative
- G contains an identity element e, i.e. for all a∈G, a∗e=e∗a=a
- For all a∈G, there exists a′∈G with a∗a′=a′∗a=e
Theorems:
- In any group (G,∗), for each a∈G, there is only one element a′∈G with a∗a′=a′∗a=e, where e is the identity element.
Problems:
Section 3
Section 4
1)Not a group. (Z,∗) fails to have an inverse at 0.
5)Not a group. Ordinary division is not associative.
7)(Z1000,+1000)
12)Not a group. Any matrices with a zero along the diagonal fail to have an inverse.
13)Is a group. The multiplication of two n×n, matrix multiplication is associative, the Identity Matrix is an identity, and any such matrices will have non-zero determinants, meaning they will have inverses.
17)Is a group. Multiplying two n×n upper-triangular matrices with determinant 1 gives another n×n upper-triangular matrix with determinant 1. Matrix multiplication is associative. The Identity Matrix is upper-trianguler and has determinant one, and the inverse of an n×n upper-triangular matrix with determinant 1 will also be an n×n upper-triangular matrix with determinant 1.