MATB24H3 Study Guide - Final Guide: Scalar Multiplication, Additive Inverse, Multiplication Table

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Published on 21 Apr 2013
School
UTSC
Department
Mathematics
Course
MATB24H3
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B24S Fall 2011
Lecture 1
1 Notation
A set is simply a collection of objects. We will use among other sets the
following:
R: the set of real numbers
Z: the set of integers (positive, negative, or zero).
Q: the set of all rational numbers= {p
q
p, q Z, q 6= 0}
Zn: the set of integers from 0 to n1 inclusive. For example Z3={0,1,2}
2 Fields
Definition :A field Fis a set of elements with two operations (called addi-
tion) and (called multiplication) satisfying the following: for all a,band c
in F, the following laws hold:
1. Fis closed under and , i.e. aband abare in Fwhenever a
and bare in F.
2. Commutative laws: ab=baand ab=ba.
3. Associative laws: (ab)c=a(bc) and (ab)c=a(bc).
4. Distributive law: a(bc) = (ab)(ac)
1
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Document Summary

A set is simply a collection of objects. We will use among other sets the following: Z : the set of integers (positive, negative, or zero). Q : the set of all rational numbers= { p q (cid:12) (cid:12) (cid:12) (cid:12) p, q z, q 6= 0} Zn : the set of integers from 0 to n 1 inclusive. Note: it is important not to assume anything about newly de ned sets with addition and multiplication. It is never assumed that the identities ex- ist. They must be found and they may not be what seem to be the obvious choices from the elements of the given set. Here are some examples of elds: the set of the real numbers with the usual addition and multiplication. Verify this result as an exercise: the set of all rational numbers with the usual addition and scalar multiplication, the set z3 with the addition and multiplication given in the following tables:

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