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University of Toronto Scarborough

Mathematics

MATB24H3

Sophie Chrysostomou

Winter

Description

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 ∈ Z,q ▯= 0}
q
Zn: the set of integers from 0 to n − 1 inclusive. For exam3le Z = {0,1,2}
2 Fields
Deﬁnition :A ﬁeld F is a set of elements with two operations ⊕ (called addi-
tion) and ⊗ (called multiplication) satisfying the following: for all a, b and c
in F, the following laws hold:
1. F is closed under ⊕ and ⊗, i.e. a ⊕ b and a ⊗ b are in F whenever a
and b are in F.
2. Commutative laws: a ⊕ b = b ⊕ a and 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 5. There exists an element in F called the additive identity or the zero
element denoted by z such that x ⊕ z = x for all x in F.
6. For any a in F, there is an additive inverse element denoted by −a in
F such that a ⊕ (−a) = z.
7. There exists an element in F called the multiplicative identity or the
identity element denoted by e such that x ⊗ e = x for all x in F.
8. For any a ▯= z in F, there exists a multiplicative inverse element denoted
−1 −1
by a in F such that a⊗a = e where e is the multiplicative identity
deﬁned above.
Note: It is important not to assume anything about newly deﬁned 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 ﬁelds:
1. The set of the real numbers with the usual addition and multiplication.
Verify this result as an exercise.
2. The set of all rational numbers with the usual addition and scalar
multiplication.
3. The set Z with the addition and multiplication given in the following
3
tables:
⊕ 0 1 2
0 0 1 2
1 1 2 0
2 2 0 1
⊗ 0 1 2
0 0 0 0
1 0 1 2
2 0 2 1
2 Note that the result of addition and scalar multiplication is the remain-
der after division by 3. For example: 2+2 = 4 and 4÷3 has remainder
1. Therefore 2 ⊕ 2 = 1
(a) Note also that

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