Math 360, Fall 2013, Assignment 3
We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads.
- - Voltaire
Carefully define the following terms, then give one example and one non-example of each:
- Isomorphism.
- Isomorphic.
- Structural property.
- Identity element.
- Group.
- Inverse element.
- Abelian group.
Carefully state the following theorems (you need not prove them):
- Uniqueness of identity element.
- Left and right cancellation laws.
Solve the following problems:
- Section 3, problems 2, 3, 4, 8, 9, 10, and 17.
- Section 4, problems 3, 4, 5, 6, 10, 11, 12, and 13.
Questions:
- Directions from 2,3, and 4:
There is a binary operation \(*\) defined on the set \(S=\{a,b,c,d,e\}\), and computed with the table:
| * | a | b | c | d | e |
| a | a | b | c | b | d |
| b | b | c | a | e | c |
| c | c | a | b | b | a |
| d | b | e | b | e | d |
| e | d | b | a | d | c |
- 2.2
Compute \((a*b)*c\) and \(a*(b*c)</math. Can you say on the basis of this computation whether '"`UNIQ--math-00000000-QINU`"'a^{-1}\in A</math> such that</p><p>\(a*a^{-1} = i = a^{-1}*a\)
Example:
-5 is the inverse of 5, over addition of integers.
Non-Example:
-5 is not the inverse of 5 over multiplication of rational numbers. 1/5 is.
- Abelian Group
Definition:
An abelian group is a group that is also commutative.
Example:
\((\mathbb{Z},+)\) is an abelian group (because addition of integers is associative, it commutes, there is an identity (0), and every element has an inverse (the negative of that element).
Non-Example:
Multiplication of invertible matrices is not an abelian group, because matrix multiplication does not commute.
Theorems:
- Uniqueness of Identity Element
If a binary structure \((A,*)\) has an identity element \(i\), then that element is unique. Meaning there are no other elements that are also identities. Formally, if there are two elements \(i,j\in A\), and \((a*i=i*a=a)\) and \( (a*j = j*a = j)\), then \( i=j\).
- Left and Right Cancellation Laws
Given elements \(x,y,z\) in a group \((G,*)\), \(x*z = y*z \rightarrow x=y</math, and <math>z*x = z*y \rightarrow x=y\). You can cancel equal elements from both sides of an equation, if they're being used on the same side of an operation. (\(x*z=z*y\) does not imply that \(x=y\) unless the group is commutative/abelian.
Questions:
- 2.2
Answer:
- 2.3
Answer:
- 2.4
Answer:
- 2.8
Answer:
- 2.9
Answer:
- 2.10
Answer:
- 2.17
Answer:
- 3.3
Answer:
- 3.4
Answer:
- 3.5
Answer:
- 3.6
Answer:
- 3.10
Answer:
- 3.11
Answer:
- 3.12
Answer:
- 3.13
Answer:
