# 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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