# Answer the following completely: 1. Determine whether each is a subset of the complex numbers is a subgroup of group c of complex numbers under ADDITION a. `Q^+` `b. 7Z` c. The set iR of pure...

Answer the following completely:

1. Determine whether each is a subset of the complex numbers is a subgroup of group c of complex numbers under ADDITION

a. `Q^+`

`b. 7Z`

c. The set *i*R of pure imaginary numbers including 0.

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### 1 Answer

To show that an infinite subset H of a group G forms a subgroup under the operation + we need only show that `a,b in H => a+b, a+(-b) in H`

(1) Determine if `Q^(+)` is a subgroup of `CC` :

Let `a,b in Q^(+)` ( e.g. a and b are positive rational numbers.)

Suppose `b>a` . Then `a+b in Q^(+)` but `a+(-b) notin Q^(+)` since `a+(-b)<0` .

**So `Q^(+)` is not a subgroup of `CC` under addition.**

** Another way to look at this is that every element in `Q^(+)` does not have an additive inverse. **

(2) Determine if `7ZZ` is a subgroup of `CC` under addition:

Let `a,b in 7ZZ` (e.g. a and b are intger multiples of 7).

(a) `a+b in 7ZZ` : let a=7m, b=7n with `m,n in ZZ` . Then

a+b=7m+7n=7(m+n) and `7(m+n) in 7ZZ`

(b) `a+(-b) in 7ZZ` : again let a=7m, b=7n with m,n integers. Then a+(-b)=7m-7n=7(m-n) and `7(m-n) in 7ZZ`

**Therefore `7ZZ` is a subgroup of `CC` under addition.**

(3) Determine if `iRR` is a subset of `CC` under addition:

Let `a,b in iRR` (e.g. a and b are pure imaginary numbers).

(a) `a+b in iRR` : Let a=ci and b=di where `i=sqrt(-1);c,d in RR` .

a+b=ci+di=(c+d)i and `(c+d)i in iRR`

(b) `a+(-b) in iRR` : Again let a=ci,b=di with i the imaginary number and c,d real numbers. Then

a+(-b)=ci-di=(c-d)i and `(c-d)i in iRR`

Therefore `iRR` is a subset of `CC` under addition.