Question

Let H = {a + bi |a,b are in R,ab≥0}. Prove or disprove that H is a subgroup of C under addition.

Answer 1

Given: H = {a + bi |a,b are in R,ab≥0}.

To find: Prove or disprove that H is a subgroup of C under addition.

Solution:

• Now we have given the set H = {a + bi |a,b are in R,ab≥0}.

• Let x = a + bi

                 x = 1 + 0i ∈ H

                 since 1.0 ≥ 0

                 x = 0 - 1i ∈ H

                 since 0(-1) ≥ 0

• But x + y = 1 - 1i ∉ H

                 as 1(-i) = -i < 0

Answer:

               So therefore, H fails to be closed with respect to addition and is not a subgroup of C.