Logical functions

Logical variable

A logical variable is a mathematical quantity that can take one of two values: true or false.

The value true is represented by the value 1.
The value false is represented by the value 0.

Logical function

A logical function applies mathematical operations to logical variables.

The value of a logical function is a logical variable.

A logical function is defined by its truth table.

NOT function

The NOT function is applied to a single variable.

This function is also called negation.

Notation
The NOT applied to the variable a is noted a.

Truth table

OR function

The OR function is applied to two variables.

This function is also called disjuntion.

Notation
The OR function applied to two variables a and b is noted a + b
and it is also noted a ∨ b.

Truth table

AND function

The AND function is applied to two variables.

This function is also called conjuntion.

Notation
The AND function applied to two variables a and b is noted a . b
and it is also noted a ∧ b.

Truth table

XOR function

The XOR function (exclusive or) is applied to two variables.

Notation
The XOR function applied to two variables a and b is noted and defined as follows:
a ⊻ b = a ⊕ b = a.b + a.b

Truth table

Properties

The logical functions OR and AND are commutative and associative.

a.(b + c) = a.b + a.c

a + (b.c) = (a+b).(a+c)

De Morgan’s laws

a + b = a . b

a.b = a + b

Example

Consider applying for a job competition.

Acceptance of the application is “TRUE” if the candidate holds a Bachelor’s degree AND has four years of professional experience, OR if the candidate holds a Master’s degree.

We have 3 variables:
a = “hold a Bachelor’s degree”
b = “have four years of professional experience”
c = “hold a Master’s degree”

The variable
d = “application accepted”
is a logical function of the variables a, b and c.

d = f(a,b,c) = a.b + c

Truth table