Determine by checking the group axiom (Associativity) whether this set is a group: (C, o), where x o y = x + y - 2i

C = set of all imaginary numbers

Let x, y, z ∈ C, the set of complex numbers.

By definition of ○,

(x ○ y) ○ z = (x + y - 2i) ○ z

… = (x + y - 2i) + z - 2i

… = x + y + z - 4i

and

x ○ (y ○ z) = x ○ (y + z - 2i)

… = x + (y + z - 2i) - 2i

… = x + y + z - 4i

so (x ○ y) ○ z = x ○ (y ○ z), which means ○ is indeed associative under C.

step-by-step explanation:

first, eliminate the parenthesis:

3

x

+

2

−

4

x

second, group like terms:

3

x

−

4

x

+

2

third, combine like terms:

(

3

−

4

)

x

+

2

→

−

1

x

+

2

→

−

x

+

2