Principles of Mathematics in Operations Research

9.2 Fields

Claim (ii): a is the greatest of the lower bounds.

Proof (ii): Show if a -< /?, /3 £ 5 => /? is not a lower bound of B.

ft fi L (because a -< (3); i.e. /? is not a lower bound of .B.

Therefore, a = inf B. •

9.2 Fields

Let us repeat Definition 2.1.1 for the sake of completeness.

Definition 9.2.1 A field is a set F / 0 with two operations, addition(+) and

multiplication(.), which satisfy the following axioms:

(A) Addition Axioms:

(Al) Vx, y £ F, x + y £ F (closed under +)

(A2) Vx,y £ F, x + y = y + x (commutative)

(A3) Vx, y,z £ F, (x + y) + z — x + (y + z) (associative)

(A4) 30 e F 9 Vx G F x + 0 = x (existence of ZERO element)

(A5) \fx £ F, 3 an element -x £ F B x + (-x) = 0 (existence of INVERSE

element)

(M) Multiplication Axioms:

(Ml) Vx,y £ F, x • y € F (closed under •)

(M2) \/x,y £ F, x • y = y • x (commutative)

(MS) Vx, y,z £ F, (x • y) • z — x • (y • z) (associative)

(M4) 31 7^ 0 3 Vx £ F, 1-x = x (existence of UNIT element)

(M5) Vx ^ 0 3 an element ^ g F 3 ji = 1 (existence of INVERSE element)

(D) Distributive Law:

Vx, y,z £ F, x • (y + z) = xy + xz

Notation :

x + (-«/) =x-y; x(-) = -; x + {y + z) = (x + y) + z

\VJ V

x • x = x^2 ; x + x = 2x; x(yz) = xyz, • • •

Example 9.2.2 F = Q with usual + and • is a field.

Example 9.2.3 Let F = {a, b, c} where a^b, a ^ c, b ^ c.

a

b

c

a b c

a a a

a b c

a c b

+

a

b

c

a b c

a b c

b c a

cab

F is a field with 0 = a, 1 — b.