Understanding the Homomorphism and Isomorphism of the Map f: ? → ?? Defined by f(x) 6?

In the realm of mathematics, particularly in abstract algebra, the concept of homomorphisms plays a crucial role in understanding the structure-preserving mappings between algebraic structures. Let's delve into the detailed analysis of the map f: ? → ?? defined by the function f(x) 6x, where ? represents the set of real numbers under addition, and ?? represents the set of positive real numbers under multiplication. This article will explore whether this map is a homomorphism and, even further, whether it is an isomorphism.

Homomorphism Definition

A map f: A → B is a homomorphism if it preserves the group operation. That is, for all a, b ∈ A, we must have f(a b) f(a) ? f(b).

The Groups in Question

- The group ? is the additive group of real numbers with the operation of addition.

- The group ?? is the multiplicative group of positive real numbers with the operation of multiplication.

Checking for Homomorphism

To check if the map f(x) 6x is a homomorphism, we need to verify if the following condition holds:

f(x y) f(x) ? f(y)

Calculations

- Calculate f(x y):

f(x y) 6x y

- Use the property of exponents: 6x y 6x ? 6y

- Calculate f(x) ? f(y):

f(x) ? f(y) 6x ? 6y

- Comparison:

We find that:

6x y 6x ? 6y

Since both sides of the equation are equal, we conclude that f(x y) f(x) ? f(y), confirming that f is a homomorphism.

Further Analysis: Isomorphism

Not only is f a homomorphism, but it also happens to be an isomorphism. An isomorphism is a bijective (one-to-one and onto) homomorphism, which means it not only preserves the group operation but also the structure of the groups.

To prove that f is an isomorphism, we need to show two things:

One-to-One (Injective): There are no two elements in ? that map to the same element in ??. Onto (Surjective): Every element in ?? is the image of some element in ?.

One-to-One Property

Say we have two elements x, y ∈ ? such that:

f(x) f(y)

6x 6y

This implies:

x y

Hence, f is one-to-one.

Onto Property

For any w ∈ ??, we can find some x ∈ ? such that:

f(x) w

This reduces to finding:

6x w

Thus:

x log? w

Since any positive real number w can be expressed as 6x for some x ∈ ?, the function f is onto.

Isomorphism as the Composition of Two Isomorphisms

The map f(x) 6x can also be viewed as a composition of two isomorphisms. Let g(x) ex, which is known to be an isomorphism. Define t ln 6. Then we have:

f(x) 6x e(ln 6)x etx g(tx)

Since the multiplication by t is an isomorphism, the composition of two isomorphisms is also an isomorphism.

Conclusion

From the above analysis, we have concluded that the map f(x) 6x is both a homomorphism and an isomorphism from the additive group ? to the multiplicative group ??.