Truth Tables

Truth tables are a way to define logical connectives $f:\{0,1{\}}^n\to \{0,1\}$ by listing all possible inputs and outputs in a table. We first assign $n$ labels $x_1,\dots ,x_n$ to the each element of the input tuple from $\{0,1{\}}^n$. The first row of the table contains these labels and the label $f(x_1,\dots ,x_n)$. Each subsequent row contains a combination of possible values for the inputs $x_1,\dots ,x_n$ and the value of the function when evaluated at those inputs.

Example: Negation

Consider the negation function $f:\{0,1\}\to \{0,1\}$ defined as

$$f(x)=\{\begin{array}{l}0\text{if }x=1\\ 1\text{if }x=0\end{array}$$

Its truth table is the following:

$x$	$f(x)$
$0$	$1$
$1$	$0$

Example: Conjunction

The conjunction function $f:\{0,1{\}}^2\to \{0,1\}$ is defined as

$$f(x,y)=\{\begin{array}{l}1\text{if and only if }x=y=1\\ 0\text{otherwise}\end{array}$$

Its truth table is the following:

$x$	$y$	$f(x,y)$
$0$	$0$	$0$
$0$	$1$	$0$
$1$	$0$	$0$
$1$	$1$	$1$