Question

There are four different functions \(f:\\{a, b\\} \rightarrow\\{0,1\\} .\) List them. Diagrams suffice.

Short answer

The four functions are: 1) \((a, 0), (b, 0)\) 2) \((a, 1), (b, 0)\) 3) \((a, 0), (b, 1)\) 4) \((a, 1), (b, 1)\). This represents all possible mappings from the domain \{a, b\} to the codomain \{0,1\}.

Step-by-step solution

Understanding Doman and Codomain

First, understand that the domain of the function is \{a, b\} and the codomain is \{0, 1\}. A function is a relation from a set of inputs (the domain) to a set of possible outputs (the codomain). Each input is related to exactly one output.

List Functions with Diagrams

There are 2 elements in the domain and each one has 2 choices of where it can map to in the codomain. Thus, there are \(2^2 = 4\) possible functions. Here are the diagrams representing each function: 1) \((a, 0), (b, 0)\) 2) \((a, 1), (b, 0)\) 3) \((a, 0), (b, 1)\) 4) \((a, 1), (b, 1)\). Each pair represents what element from our domain maps to what element in our codomain.