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Example: Examples of Boolean Functions

(related to Lemma: Boolean Function)

Example 1

The Boolean constants $1$ and $0$ are particular types of prime propositions. They define constant Boolean functions $f(1)=1$ and $f(0)=0$.

Example 2:

A Boolean variable $x$ is another particular type of a prime proposition. They define an identity Boolean functions $$f(x)=x=\cases{0&\text{if }x=0,\\1&\text{if }x=1.}$$

Example 3

Let \(f(x_1,x_2)\) be represented by the compound proposition \(x_1\vee \neg x_2\). Note that this proposition involves two connectives - the negation "$\neg$", which is an unary connective and the disjunction "$\vee$", which is a binary connective. The compound propositions involves two variables $x_1$ and $x_2$, which are prime propositions.

Using an interpretation $I$ and the corresponding valuation function $[[]]_I$ we can calculate the values of the Boolean function $f$ by assigning all possible truth values to the two variables of $f$:

• \(f(1,1)=[[1\vee \neg 1]]_I=[[1\vee 0]]_I=[[ 1 ]]_I=1,\)
• \(f(1,0)=[[1\vee \neg 0]]_I=[[1\vee 1]]_I=[[ 1 ]]_I=1,\)
• \(f(0,1)=[[0\vee \neg 1]]_I=[[0\vee 0]]_I=[[ 0 ]]_I=0,\)
• \(f(0,0)=[[0\vee \neg 0]]_I=[[0\vee 1]]_I=[[ 1 ]]_I=1.\)

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