Let X be a field and let x y z in X Prove a The multiplica

Let (X, +, .) be a field and let x, y, z in X.

Prove a) The multiplicative identity is unique.

b) If xz = yz and z is not equal to 0, then x = y

c) If x is not equal to 0, then (x^-1)^-1= x

Solution

b) If xz = yz and z is not equal to 0, then x = y

xz = yz

xz - yz = 0

(x - y)z = 0

so x = y or z = 0

but given that z is not equal to zero

so x = y (proved)

c) If x is not equal to 0, then (x^-1)^-1= x

(x^-1)^-1 = 1/ x^-1 = x^+1 = x (proved)

a) The multiplicative identity is unique.

Suppose that 2\' in F satisfies that 2\' a = a for each a in F

choose a = 1

2\' 1 = 1

Since (F3)

1 a = a for each a in F

choose a = 2\'

1 2\' = 2\'

By (F1), it fallows that

2\' = 1 2\' = 2\' 1 = 2

So the multiplicative identity is unique.