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MAT425 Real Analysis L3

Prof. Thistleton



Countable and Uncountable Sets (continued)


Definitions


Theorem: Q is countable.


Uncountable Sets and Cantor Diagonalization


Infinite Intersections and Infinite Unions

Let {\(S_{\alpha} : \alpha \in A\)} be a collection of sets where A is some (possible infinite) index set.

1.
Union
















































2.
Intersection


Limit Inferior and Limit Superior

1.
lim inf
























2.
lim sup


Fields


We can try to understand what the real numbers are by first understanding how they behave. Abstract algebra is an area of mathematics that looks at sets and operations like addition or multiplication defined on these sets to see what the properties of these systems are.

For the real numbers, we believe the following statements to be true and have worked with these properties for years. We will consider these to be axioms and define a field to be a set together with operations called addition and multiplication for which the following statements are true. Note that the operations we define on a set might not be the same "good old" addition and multiplication to which you are accustomed.

1.
Closure under addition and subtraction: If we add or multiply two elements of a field we obtain an element of the field.

2.
Addition is associative: If x, y, and z are three elements from the field then
(x+y)+z = x+(y+z).

3.
Addition is commutative: If x and y are elements of the field then x+y = y+x.

4.
There is a field element, called the zero element and denoted , such that, for any x in the field, 0+x=x.

5.
For every field element x there is a field element y called the additive inverse of x such that x + y = 0. We shall denote the additive inverse of x as -x.

6.
Multiplication is both associative and commutative.

7.
There is a field element called the multiplicative identity and denoted as 1, such that \(1\cdot x=x\) for every x in the field.

8.
For each x in the field (except for the zero element) there is a field element y called the multiplicative inverse of x such that \(x\cdot y = 1\). Denote this element y as x-1.

9.
For any x, y, z in the field, \(x\cdot (y+z) = x\cdot y + x\cdot z\). We say that multiplication distributes over addition.


Is the set of natural numbers, together with the usual addition and multiplication, a field?











Is the set of integers, together with the usual addition and multiplication, a field?











Is the set of rational numbers, together with the usual addition and multiplication, a field?











Is the set of real numbers, together with the usual addition and multiplication, a field?











Is the set of complex numbers a field? How are addition and multiplication defined on the complex numbers?


Show that in a field the additive inverse is unique.














Show that in a field each nonzero element has a unique multiplicative inverse.














Denote the additive inverse of x as -x. Is it true that \(-x = (-1)\cdot x\)? (What do we mean by -1?)














Show that in a field \(x \cdot y = 0\ \ \Rightarrow x=0\ or\ y=0\).


We can define modular arithmetic (clock arithmetic) on the set of integers \(0, 1, \cdots, 11\). Define addition on this set as follows: For any two elements \(x,\ y \in \{0,1,\cdots,11\}\) define \(x \oplus y\) as the remainder of (x + y)/12. For example, \(3 \oplus 4 = 7(mod\ 12)\), \(6 \oplus 6 = 0(mod\ 12)\)(why?) and \(10 \oplus 8 = 6(mod\ 12)\).








Multiplication is defined similarly. To evaluate \(x \otimes y\)we take the appropriate multiple of 12 away from \(x \cdot y\) so that the remainder is an element of \(\{0,1,\cdots,11\}\). For example, \(4 \otimes 2 = 8(mod\ 12)\), \(3 \otimes 4 = 0(mod\ 12)\), and \(11 \otimes 11 = 1(mod\ 12)\).








Is this a field?


























Can we take \(\{0,1,\cdots,n\}\) with clock arithmetic to generate a field for any value of n?


 
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William Thistleton
11/10/1998