The set of Complex numbers (C) with addition:

Has an identity element, but does not follow the other conditions for a group.

Has inverses, but does not follow the other conditions for a group.

Is a group.

Is closed under the operation, but does not follow the other conditions for a group.

Option C

A complex number can be written in the form a + bi where a and b are Real numbers and

i = √(-1)

• If a + bi ∈ C, (a + bi) + (0 + 0i) = a + bi is always true, so (0 + 0i) is the identity element.

• If a + bi ∈ C, (a + bi) + (-a − bi) = 0 + 0i and (-a − bi) + (a + bi) = 0 + 0i (the identity element), so (a + bi) and (-a − bi) are inverses of each other.

• For any three complex numbers (a_{1} + b_{1}i), (a_{2} + b_{2}i) and (a_{3} + b_{3}i) in C:

(a_{1} + b_{1}i) + [(a_{2} + b_{2}i) + (a_{3} + b_{3}i)] = [(a_{1} + b_{1}i) + (a_{2} + b_{2}i)] + (a_{3} + b_{3}i)

⇒ The associative law has been followed.

• If (a_{1} + b_{1}i) and (a_{2} + b_{2}i) are any two Complex numbers:

(a_{1} + b_{1}i) + (a_{2} + b_{2}i) = (a_{1} + a_{2}) + (b_{1} + b_{2})i, which is also a Complex number.

⇒ The closure law has been followed.

Therefore, all four conditions for a group are followed by the set C under addition.

⇒ C is a group with respect to addition.

- 2024-06-12 03:15
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- 25 Full Marks
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