If-Then-Else

boolean-algebra

Definition

If-Then-Else

The If-Then-Else (ITE) function is a Boolean function that is defined as:

$$\text{ITE}(a,b,c):⟺(a\wedge b)\vee (\wedge \neg a\wedge c)$$

Example

For example, the following equation can be represented as code:

$$(a\wedge \neg b)\vee \neg a$$

if (a == true) then
	if (b == true) then
		false
	else
		true
else
	true

Using the $\text{ITE}$ function, this pseudo-code can be naively represented as the following:

$$\text{ITE}(a,\text{ITE}(b,false,true),true)$$

Of course, this can be simplified to:

$$\text{ITE}(a,\text{ITE}(b,false,true),true)=\text{ITE}(a,\neg b,true)$$