Boolean elements

Whether creating equations in Boolean algebra or using them in your programs, you’ll form both simple and complex logical expressions that use basic operations to combine the logical conditions.

Notation

Boolean (logical) equations are expressed in a way similar to mathmatical equations. Variables in Boolean expressions though, have only two possible values, true or false. For an equation using a logical expression, the equivalant sides of the equal sign ,=, will be only true or false too.

The following list shows the basic notation elements for Boolean expressions.

• ~A: the inverse (NOT) of A, when A is true, ~A is false
• A + B: the value of A OR B
• A · B: the value of A AND B
• A ⊕ B: the value of the exclusive OR (XOR) of A with B
• Q: equivalent result (OUTPUT) value of a logical expression

A resulting value, Q, from a logical expression in is shown like:

Q = A + B

An equation to show logically equivalent expressions (where both sides have the same resulting value) can look like this:

~(A + B) = ~A · ~B

Logical operators

All Boolean expressions result from a combination of conditions and operators. These operators join individual conditons together and evaluate into a single true or false condition. The following are the basic logical operators. Their use in both Boolean algebra and in code is shown along with their truth table.

Identity

Identity means that a result value is the same as the condition itself.

Q = A

let A = false
let Q = A

Example - Blink pixels on press

let A = false
forever(function () {
    A = input.buttonA.isPressed()
    if (A) {
        light.setAll(0xff0000)
    } else {
        light.clear()
    }
    pause(500)
})

Truth table

NOT (Negation)

The NOT operator is called negation or the inverse. It takes a single logical value and makes it have the opposite value, true goes to false and false goes to true.

Q = ~A

let A = false
let Q = !(A)

Example - Blink pixels on not pressed

let A = false
forever(function () {
    A = input.buttonA.isPressed()
    if (!(A)) {
        light.setAll(0xff0000)
    } else {
        light.clear()
    }
    pause(500)
})

Truth table

OR (Disjunction)

The OR operator results in true when one or more conditions are true.

Q = A + B

let A = false
let B = false
let Q = A || B

Example - Blink on any press

let A = false
let B = false
forever(function () {
    A = input.buttonA.isPressed()
    B = input.buttonB.isPressed()
    if (A || B) {
        light.setAll(0xff0000)
    } else {
        light.clear()
    }
    pause(500)
})

Truth table

AND (Conjunction)

The AND operator requires that all conditions are true for the result to be true.

Q = A · B

let A = false
let B = false
let Q = A && B

Example - Blink on double press only

let A = false
let B = false
forever(function () {
    A = input.buttonA.isPressed()
    B = input.buttonB.isPressed()
    if (A && B) {
        light.setAll(0xff0000)
    } else {
        light.clear()
    }
    pause(500)
})

Truth table

XOR (Exclusive OR)

Exclusive OR (XOR) means that only one or the other condition is true. Both conditions can’t be true at the same time. XOR is common in Boolean algebra but it has no operator in JavaScript. Its operation can be made from combining a few simple expressions.

Q = A ⊕ B

let A = false
let B = false
let Q = (A || B) && !(A && B)

Example - Blink on one press or the other

let A = false
let B = false
forever(function () {
    A = input.buttonA.isPressed()
    B = input.buttonB.isPressed()
    if ((A || B) && !(A && B)) {
        light.setAll(0xff0000)
    } else {
        light.clear()
    }
    pause(500)
})

Truth table