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

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Published on 21 Apr 2013

Department

Mathematics

Course

MATB24H3

Professor

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 n−1 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. a⊕band a⊗bare in Fwhenever a

and bare in F.

2. Commutative laws: a⊕b=b⊕aand a⊗b=b⊗a.

3. Associative laws: (a⊕b)⊕c=a⊕(b⊕c) and (a⊗b)⊗c=a⊗(b⊗c).

4. Distributive law: a⊗(b⊕c) = (a⊗b)⊕(a⊗c)

1

## 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: