Question

Suppose \(A=\\{a, b, c\\} .\) Let \(f: A \rightarrow A\) be the function \(f=\\{(a, c),(b, c),(c, c)\\},\) and let \(g: A \rightarrow A\) be the function \(g=\\{(a, a),(b, b),(c, a)\\} .\) Find \(g \circ f\) and \(f \circ g\).

Short answer

\(g \circ f = \{(a, a), (b, a), (c, a)\}\) and \(f \circ g = \{(a, c), (b, c), (c, c)\}\)

Step-by-step solution

- Find \(g \circ f\)

First, we find \(g \circ f\). This composition means, first apply \(f\) and then apply \(g\) to the result. From \(f = \{(a, c), (b, c), (c, c)\}\), we can see that \(f\) maps every element of \(A\) to \(c\). So \(g(f(a)) = g(f(b)) = g(f(c)) = g(c) = a\). Thus, \(g \circ f = \{(a, a), (b, a), (c, a)\}\).

- Find \(f \circ g\)

Next, we calculate \(f \circ g\), which means, first apply \(g\) and then apply \(f\) to the result. From \(g = \{(a, a), (b, b), (c, a)\}\), we see that \(g(a) = a\), \(g(b) = b\), and \(g(c) = a\). So, \(f(g(a)) = f(a) = c\), \(f(g(b)) = f(b) = c\), and \(f(g(c)) = f(a) = c\). Therefore, \(f \circ g = \{(a, c), (b, c), (c, c)\}\).