Notation of a boolean function

Question

I'm studying Boolean algebra but I was confused as the notation of a Boolean function. When I write/denote a Boolean function that way, what does that mean? $$ f: \mathbb{Z}^2_2 \rightarrow\mathbb{Z}_2 $$

Answer 1

It's a function mapping pairs $(a, b) \in \mathbb{Z}_{2} \times \mathbb{Z}_{2}$ into elements of $\mathbb{Z}_{2}$. Note that $\mathbb{Z}_{2}$ is the set of elements $\{0, 1\}$ with the operations of addition as exclusive-or, and multiplication as you would expect.

Note: $\mathbb{Z}_{2}^{2} \cong \mathbb{Z}_{2} \times \mathbb{Z}_{2}$.

Answer 2

$\Bbb Z$ is the set of Integers.

$\Bbb Z_2$ is a set of two integers, specifically: $\{0, 1\}$, also called the Booleans or Boolean numbers.

$\Bbb Z_2^2$ is the set of ordered pairs (or 2-tuple) of Boolean numbers; the square symbol means take the Cartesian product of the set with itself: $$\begin{align}\Bbb Z_2^2 & = \Bbb Z_2\times \Bbb Z_2 \\ & = \{0,1\}\times\{0,1\} \\ & = \{(0,0), (0,1), (1,0), (1,1)\}\end{align}$$

$f:\Bbb Z_2^2\to \Bbb Z$ is a function which maps a 2-tuple of Boolean values (the arguments) into a Boolean value (the result).

For example, we can define the or-function: $f_\vee : \Bbb Z_2^2\to \Bbb Z$ as $$\begin{align}f_\vee(x,y) \;& \mathop{:=} \begin{cases} 1 & : (x,y)\in\{(0,1), (1,0), (1,1)\} \\ 0 & : (x,y)\in\{(0,0)\}\end{cases} \\[1ex] & = x\vee y\end{align}$$