Question

let G be the group (R+,×) of positive real numbers under multiplication and let H be the additive group R then the mapping given by f(x)=log x is an isomorphism​

Answer 1

Answer:

Let C6 = 〈a〉 = {1, a, a2,a3,a4,a5} and C4 = 〈b〉 = {1, b, b2,b3} where |a| = ... Indeed, if x = α(a) ∈ C4 is known then α(ai) = α(a)i = xi for ... Are the additive groups Z and Q isomorphic?

Answer 2

Answer:

therefore the function f  is surjective and hence an isomorphism

Step-by-step explanation:

We see that for r , q ∈ R,

f(r+q)=e^(r+q)=e^r . e^q=f(r) ∗ f(q)) as well as (f(0) = e^0 = 1 = identity)

So f is a group which belongs to homomorphism.

for 1-1 (injectivity)

Let r, q in Q  be placed such that f(r) = f(q). Then

(e^r=e^q)  so, (e^r/e^q=1)

or, (e^(r−q)=1)

,so r - 1 = 0 . hence, r=q and f is injective.

for Surjectivity,

for any p in R+, we see that

(ln(p)) is defined, so

(f(ln(p)) = e^ln(p) = p)

therefore the function f  is surjective and hence an isomorphism.

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