Total number of function homomorphism from Z to Q:

A function homomorphism is a special type of function that preserves the structure between two algebraic systems. In this case, we are looking for function homomorphisms from the set of integers (Z) to the set of rational numbers (Q).

Definition of a function homomorphism:

A function homomorphism between two algebraic structures A and B is a function f: A -> B that preserves the operations between the elements of A and B. In other words, for any two elements a and b in A, the homomorphism satisfies the following properties:

1. f(a + b) = f(a) + f(b)
2. f(a * b) = f(a) * f(b)

Understanding Z and Q:

- Z represents the set of integers, which includes positive and negative whole numbers, as well as zero. Example: {..., -3, -2, -1, 0, 1, 2, 3, ...}
- Q represents the set of rational numbers, which includes all numbers that can be expressed as a ratio of two integers. Example: {..., -3/2, -1/2, 0, 1/2, 3/2, ...}

Finding the total number of function homomorphisms:

To find the total number of function homomorphisms from Z to Q, we need to consider how the properties of a homomorphism are satisfied.

1. f(a + b) = f(a) + f(b):
- In Z, addition is defined as the sum of two integers.
- In Q, addition is defined as the sum of two rational numbers.
- To satisfy this property, f(a + b) must be equal to f(a) + f(b) for all a and b in Z.
- Since Q contains all possible ratios of integers, there are infinite possibilities for f(a + b), f(a), and f(b) that satisfy this property.

2. f(a * b) = f(a) * f(b):
- In Z, multiplication is defined as the product of two integers.
- In Q, multiplication is defined as the product of two rational numbers.
- To satisfy this property, f(a * b) must be equal to f(a) * f(b) for all a and b in Z.
- Similar to addition, there are infinite possibilities for f(a * b), f(a), and f(b) that satisfy this property.

Conclusion:

Due to the infinite possibilities for function homomorphisms from Z to Q, it is not possible to determine the total number of such homomorphisms. However, it is important to note that any function that preserves the operations of addition and multiplication between integers and rational numbers can be considered a function homomorphism from Z to Q.