Question

Consider the functions \(f: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}\) defined as \(f(m, n)=m+n\) and \(g: \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}\) defined as \(g(m)=(m, m)\). Find the formulas for \(g \circ f\) and \(f \circ g\).

Short answer

The formula for the composition \( g \circ f \) is \( (g \circ f) (m, n) = (m + n, m + n) \). The formula for the composition \( f \circ g \) is \( (f \circ g) (m) = 2m \).

Step-by-step solution

Composition \( g \circ f \)

Firstly, we will find the formula for the composition \( g \circ f \). The function f takes pairs of integers as input and outputs an integer, while g takes in an integer and outputs a pair of integers. As a result, the composition \( g \circ f \) will take a pair of integers as input and output a pair of integers. Indeed, for any \( (m, n) \in \mathbb{Z} \times \mathbb{Z} \):\( (g \circ f) (m, n) = g(f(m, n)) = g(m + n) = (m + n, m + n) \). Therefore, the formula for \( g \circ f \) is \( (g \circ f) (m, n) = (m + n, m + n) \).

Composition \( f \circ g \)

Secondly, we will find the formula for the composition \( f \circ g \). The function g takes an integer as input and outputs a pair of integers, while f takes a pair of integers and outputs an integer. As a result, the composition \( f \circ g \) will take an integer as input and output an integer. Indeed, for any \( m \in \mathbb{Z} \): \( (f \circ g) (m) = f(g(m)) = f(m, m) = m + m = 2m \). Therefore, the formula for \( f \circ g \) is \( (f \circ g) (m) = 2m \).