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Boolean Algebra

Boolean algebra (or less commonly symbolic logic) is a branch of algebra that deals with only two logic values - 0 (corresponding to false) and 1 (corresponding to true).

Today, Boolean algebra is the primary mathematical tool used in designing modern digital systems. Switching functions are described using Boolean algebra since they deal with two discrete states - ON and OFF (or 1 and 0). Those functions are in turn implemented via transistors which act as switches, a natural implementation for representing Boolean algebra operations. Once primitive Boolean operation circuits such as NOT, AND, and OR gates are implemented, any conceivable system of logic can be implemented using them like Lego pieces.

Variables[edit]

Main article: Boolean Variables and boolean data type

Boolean algebra uses variables just like normal algebra. Those variables can only have one of two values - either a 0 or a 1. Variable are commonly represented as a single alphabet letter. While there is no one acceptable convention, a it's not uncommon to see letters such as Equation upper A comma upper B comma and upper C used for inputs and Equation upper P comma upper Q comma upper R comma and upper Z for output. That's also the convention used on WikiChip. Sometimes it's desired to represent the negated (opposite) value of a variable, that's often done with a bar or a tick (prime) above or next to the letter, for example Equation upper A overbar or Equation normal not-sign upper B although other notations exist. Equation upper A overbar is read "not A", regardless of notation.

0 & 1[edit]

Boolean algebra deals with only two unique states or values. We often represent those two values as Equation 0 and Equation 1 . However it's important to understand that those are just two convenient representations. This is often written as Equation double-struck upper B equals StartSet 0 comma 1 EndSet . You can assign Equation black medium square and Equation black up pointing small triangle instead and the math should work just fine.

Operations & Truth tables[edit]

Main article: Boolean Operations and truth table

Boolean algebra has a set of operations that can be performed on Boolean values, those operations are conveniently enough called binary operations. The three common Boolean operators are AND, OR, and NOT. Understanding those operators can better be done by examining their behavior via tool called a truth table. truth tables is a table that lists all possible input values and their respective output values. Truth tables can be said to be the unique signature of a specific Boolean function. Truth tables are an excellent way of seeing the relationships between input values and given Boolean expressions. While there may be many ways to realize or construct a Boolean function to represent a specific relation, they all share the very same truth table. A truth-vector is a truth table in vector form.

AND operator[edit]

Main article: conjunction
Inputs Outputs
A B Q
0 0 0
0 1 0
1 0 0
1 1 1

The Boolean operator AND is usually represented by either Equation logical-and , Equation dot , or no symbol at all: for example " Equation upper A logical-and upper B ", " Equation upper A dot upper B ", and " Equation upper A upper B " are all equivalent and are read "A AND B". The behavior of this operator is shown in the truth table on the right. The result of "A AND B" is true if both A and B are true; otherwise the result is false. This expression is also called a Boolean product.

For example, suppose we have the function Equation f left-parenthesis a comma b comma c right-parenthesis equals left-parenthesis a logical-and b right-parenthesis logical-and c

Equation StartLayout 1st Row 1st Column upper Q 2nd Column equals f left-parenthesis 1 comma 0 comma 1 right-parenthesis 2nd Row 1st Column Blank 2nd Column equals left-parenthesis 1 logical-and 0 right-parenthesis logical-and 1 3rd Row 1st Column Blank 2nd Column equals 0 logical-and 1 4th Row 1st Column Blank 2nd Column equals 0 Subscript 2 EndLayout

Or

Equation StartLayout 1st Row 1st Column upper Q 2nd Column equals f left-parenthesis 1 comma 1 comma 1 right-parenthesis 2nd Row 1st Column Blank 2nd Column equals left-parenthesis 1 logical-and 1 right-parenthesis logical-and 1 3rd Row 1st Column Blank 2nd Column equals 1 logical-and 1 4th Row 1st Column Blank 2nd Column equals 1 Subscript 2 EndLayout

OR operator[edit]

Main article: disjunction
Inputs Outputs
A B Q
0 0 0
0 1 1
1 0 1
1 1 1

The Boolean operator OR is usually represented by Equation logical-or or Equation plus operators. For example " Equation upper A logical-or upper B " and " Equation upper A plus upper B ". The expression Equation upper A plus upper B is read "A or B". The result of "A OR B" is true if either A is true or B is true; otherwise the result is false. This expression is also called a Boolean sum.

For example, suppose we have the function Equation f left-parenthesis a comma b comma c right-parenthesis equals left-parenthesis a logical-or b right-parenthesis logical-or c

Equation StartLayout 1st Row 1st Column upper Q 2nd Column equals f left-parenthesis 1 comma 0 comma 1 right-parenthesis 2nd Row 1st Column Blank 2nd Column equals left-parenthesis 1 logical-or 0 right-parenthesis logical-or 1 3rd Row 1st Column Blank 2nd Column equals 1 logical-or 1 4th Row 1st Column Blank 2nd Column equals 1 Subscript 2 EndLayout

Or

Equation StartLayout 1st Row 1st Column upper Q 2nd Column equals f left-parenthesis 0 comma 0 comma 1 right-parenthesis 2nd Row 1st Column Blank 2nd Column equals left-parenthesis 0 logical-or 0 right-parenthesis logical-or 1 3rd Row 1st Column Blank 2nd Column equals 0 logical-or 1 4th Row 1st Column Blank 2nd Column equals 1 Subscript 2 EndLayout

NOT operator[edit]

Main article: negation
Inputs Outputs
A Q
0 1
1 0

The Boolean operator NOT is represented by many notations, the three most popular ones are " Equation normal not-sign upper A ", " Equation upper A overbar ", and " Equation upper A prime ". Note that unlike the AND and OR operators, the NOT operator is a unary operator and is thus drawn above or on the side of the variable. The expression Equation upper A overbar is read "not A". The truth table for the NOT operator is shown on the right. The result of the NOT operator is true if A is false, otherwise the result is true. This expression is called a Boolean complement.

For example, suppose we have the function Equation f left-parenthesis a right-parenthesis equals a overbar

Equation StartLayout 1st Row 1st Column upper Q 2nd Column equals f left-parenthesis 1 right-parenthesis 2nd Row 1st Column Blank 2nd Column equals ModifyingAbove 1 With bar 3rd Row 1st Column Blank 2nd Column equals 0 Subscript 2 EndLayout

Or

Equation StartLayout 1st Row 1st Column upper Q 2nd Column equals f left-parenthesis 0 right-parenthesis 2nd Row 1st Column Blank 2nd Column equals ModifyingAbove 0 With bar 3rd Row 1st Column Blank 2nd Column equals 1 Subscript 2 EndLayout

Order of operations[edit]

Main article: Order of Operations

So far we've made it simple by explicitly using parenthesis in all of our examples to indicate a certain part of the expression is evaluated before another part. The order of operations of a Boolean expression is very important to obtain correct result. For example consider the function Equation f left-parenthesis a comma b comma c right-parenthesis equals a logical-and b logical-or c for input Equation f left-parenthesis 0 comma 0 comma 1 right-parenthesis . Does it mean Equation left-parenthesis 0 logical-and 0 right-parenthesis logical-or 1 equals left-parenthesis 0 right-parenthesis logical-or 1 equals 1 ? or does it mean Equation 0 logical-and left-parenthesis 0 logical-or 1 right-parenthesis equals 0 logical-and left-parenthesis 1 right-parenthesis equals 0 ? Same expression, different results. It turns out the the correct order is Equation left-parenthesis a logical-and b right-parenthesis logical-or c (and Equation f left-parenthesis 0 comma 0 comma 1 right-parenthesis equals 1 ). In Boolean expressions, the NOT operator has the highest precedence, followed by AND, then OR.

For example,

Equation f left-parenthesis a comma b comma c right-parenthesis equals upper A logical-and upper B overbar logical-or upper A logical-and upper C overbar logical-or upper A logical-and upper B logical-and upper C equals left-parenthesis left-parenthesis upper A logical-and left-parenthesis upper B overbar right-parenthesis right-parenthesis logical-or left-parenthesis upper A logical-and left-parenthesis upper C overbar right-parenthesis right-parenthesis logical-or left-parenthesis left-parenthesis upper A logical-and upper B right-parenthesis logical-and upper C right-parenthesis right-parenthesis

and

Equation f left-parenthesis a comma b right-parenthesis equals upper A overbar logical-or upper B logical-and upper A equals left-parenthesis left-parenthesis upper A overbar right-parenthesis logical-or left-parenthesis upper B logical-and upper A right-parenthesis right-parenthesis

Axioms[edit]

Main article: Boolean Algebra Axioms

Boolean algebra is govern by a set of special axioms that say what kind of Boolean expression manipulations can be done. They are called axioms because they not things that have to be proven but rather part of the definition of Boolean algebra. Many of those laws are common to both Boolean algebra and ordinary algebra. Using those laws, equations can be converted into different forms. One particular transformation known as minimization plays a crucial role in the design of logic circuits. One last thing to note before we get to the actual laws is that Boolean algebra identities come in pairs. This is known as duality principle and it is covered in much more detail later on.

Axiom AND form OR form
Identity Axiom Equation a logical-and 1 equals a Equation a logical-or 0 equals a
Inverse Axiom Equation a logical-and a overbar equals 0 Equation a logical-or a overbar equals 1
Commutative Axiom Equation a logical-and b equals b logical-and a Equation a logical-or b equals b logical-or a
Associative Axiom Equation left-parenthesis a logical-and b right-parenthesis logical-and c equals a logical-and left-parenthesis b logical-and c right-parenthesis Equation left-parenthesis a logical-or b right-parenthesis logical-or c equals a logical-or left-parenthesis b logical-or c right-parenthesis
Distributive Axiom Equation a logical-or left-parenthesis b logical-and c right-parenthesis equals left-parenthesis a logical-or b right-parenthesis logical-and left-parenthesis a logical-and c right-parenthesis Equation a logical-and left-parenthesis b logical-or c right-parenthesis equals left-parenthesis a logical-and b right-parenthesis logical-or left-parenthesis a logical-or c right-parenthesis


In addition to those five axioms, there are a number of other handful laws. Those laws can be proven using the axioms we've introduced above.

Law AND form OR form
Complement Law Equation ModifyingAbove 1 With bar equals 0 Equation ModifyingAbove 0 With bar equals 1
Dominance Law Equation a logical-and 0 equals 0 Equation a logical-or 1 equals 1
Idempotent Law Equation a logical-and a equals a Equation a logical-or a equals a
Absorption Law Equation a logical-and left-parenthesis a logical-or b right-parenthesis equals a Equation a logical-or left-parenthesis a logical-and b right-parenthesis equals a
DeMorgan's Law Equation ModifyingAbove left-parenthesis a logical-and b right-parenthesis With bar equals a overbar logical-or b overbar Equation ModifyingAbove left-parenthesis a logical-or b right-parenthesis With bar equals a overbar logical-and b overbar
Involution Law Equation ModifyingAbove left-parenthesis a overbar right-parenthesis With bar equals a

It's interesting to note that it's easy to see the divergence between Boolean algebra and ordinary algebra from those laws. For example consider Equation 1 plus 1 . From Dominance Law we know the answer is Equation 1 . This is clearly not true for ordinary algebra where Equation 1 plus 1 equals 2 . Likewise from the Absorption Law we know that Equation 1 plus left-parenthesis 1 dot 1 right-parenthesis equals 1 while in ordinary algebra this is not true either.

Axioms explanation[edit]

The Identity Axiom simply states that any expression ANDed with 1 or ORed with 0 results in the original expression. Identity elements or simply identities are elements that when used with their appropriate operator leave the original element unchanged. In the case of Boolean algebra, the identity element for AND is 1 and 0 for OR.

Example:

Equation StartLayout 1st Row 1st Column f left-parenthesis a right-parenthesis 2nd Column equals a logical-and 1 logical-or 0 2nd Row 1st Column Blank 2nd Column equals a logical-or 0 3rd Column By Identity Axiom 3rd Row 1st Column Blank 2nd Column equals a 3rd Column By Identity Axiom EndLayout

The Inverse Axiom simply states that when you AND or OR an expression with its complement results in the identity element for that operation.

Example:

Equation StartLayout 1st Row 1st Column f left-parenthesis a comma b right-parenthesis 2nd Column equals left-parenthesis a logical-and a overbar right-parenthesis logical-or ModifyingAbove left-parenthesis b logical-and b overbar right-parenthesis With bar 2nd Row 1st Column Blank 2nd Column equals 0 logical-or ModifyingAbove left-parenthesis b logical-and b overbar right-parenthesis With bar 3rd Column By Inverse Axiom 3rd Row 1st Column Blank 2nd Column equals 0 logical-or ModifyingAbove 0 With bar 3rd Column By Inverse Axiom 4th Row 1st Column Blank 2nd Column equals 1 3rd Column By Inverse Axiom EndLayout

The Commutative Axiom states that individual elements in an expressions can be reordered without affecting the meaning of the expression. For example Equation a logical-and b logical-and c equals c logical-and a logical-and b .


The Associative Axiom states that individual elements in an expression can be regrouped without affecting the meaning of the expression. For example Equation left-parenthesis a logical-and b right-parenthesis logical-and left-parenthesis c logical-and d right-parenthesis equals a logical-and left-parenthesis b logical-and c right-parenthesis logical-and d . Simply put, it makes no difference in what order you group the expressions when ANDing or ORing several expressions together.


The Distributive Axiom states that ANDing distributes over over ORing. That is, ORing several expressions and ANDing the result is equaivilent to ANDing the result with a each of the individual expressions then ORing the product. Often times the Distributive Axiom is applied in reverse in a similar way to factoring in ordinary algebra. For example, given Equation left-parenthesis a logical-and b right-parenthesis logical-or left-parenthesis a logical-and c right-parenthesis , the Equation a expression can be factored out to Equation a logical-and left-parenthesis b logical-or c right-parenthesis

Canonical Forms[edit]

a b c minterms notation
0 0 0 Equation a overbar logical-and b overbar logical-and c overbar Equation m 0
0 0 1 Equation a overbar logical-and b overbar logical-and c Equation m 1
0 1 0 Equation a overbar logical-and b logical-and c overbar Equation m 2
0 1 1 Equation a overbar logical-and b logical-and c Equation m 3
1 0 0 Equation a logical-and b overbar logical-and c overbar Equation m 4
1 0 1 Equation a logical-and b overbar logical-and c Equation m 5
1 1 0 Equation a logical-and b logical-and c overbar Equation m 6
1 1 1 Equation a logical-and b logical-and c Equation m 7

Earlier we've covered truth tables which are like signatures; there are many ways to represent the same logic, however it will always result in the very same truth table. When two Boolean functions result in the same exact truth table, the two functions are said to be logically equivalent. The different representations of a truth table are known as forms. In an attempt to eliminate confusion, a few forms were were chosen to be canonical or standard forms. Before we describe those forms we need to go over a few terms.

Sum of Products (SoP)[edit]

A minterm is the Boolean product (ANDing) of Equation n variables and contains all Equation n variables of the function just once, in either normal or complemented form. For example, for the function Equation f left-parenthesis a comma b right-parenthesis with two variables, we can have the following minterms: Equation a logical-and b , Equation a logical-and b overbar , Equation a overbar logical-and b , and Equation ModifyingAbove a logical-and b With bar . If the value assigned to a variable is 0, the variable is complemented, conversely a variable remains uncomplemented if the value assigned to it is 1. Consider the table to the right, since in the first row, the variables Equation a , Equation b , and Equation c are all Equation 0 , the minterms for them is Equation a overbar logical-and b overbar logical-and c overbar

A sum term is the Boolean sum (ORing) of variables as a subset of the possible variables or their complements. For example, for the function Equation f left-parenthesis a comma b comma c right-parenthesis , the following are a few possible sum terms: Equation c overbar , Equation a logical-or b , and Equation a overbar logical-or b .

The sum of minterms also called Sum of Product (SoP), canonical sum of products, minterm expansion, and canonical disjunctive normal form (CDNF) is a Boolean expression in which each term contains all the variables, either in normal or complemented form as sum of all the minterms. For example, consider the following Boolean functions.

Equation f left-parenthesis a comma b comma c comma d right-parenthesis equals a left-parenthesis b plus c plus d right-parenthesis

We can express that function in SoP form as follows.

Equation f left-parenthesis a comma b comma c comma d right-parenthesis equals left-parenthesis a b c d right-parenthesis plus left-parenthesis a b c d overbar right-parenthesis plus left-parenthesis a b c overbar d right-parenthesis plus left-parenthesis a b c overbar d overbar right-parenthesis plus left-parenthesis a b overbar c d right-parenthesis plus left-parenthesis a b overbar c d overbar right-parenthesis plus left-parenthesis a b overbar c overbar d right-parenthesis
Sum of Product Example
Function
Equation f left-parenthesis a comma b comma c right-parenthesis equals a logical-or left-parenthesis b logical-and c right-parenthesis
a b c minterms notation Equation f Equation f prime
0 0 0 Equation a overbar logical-and b overbar logical-and c overbar Equation m 0 0 1
0 0 1 Equation a overbar logical-and b overbar logical-and c Equation m 1 0 1
0 1 0 Equation a overbar logical-and b logical-and c overbar Equation m 2 0 1
0 1 1 Equation a overbar logical-and b logical-and c Equation m 3 1 0
1 0 0 Equation a logical-and b overbar logical-and c overbar Equation m 4 1 0
1 0 1 Equation a logical-and b overbar logical-and c Equation m 5 1 0
1 1 0 Equation a logical-and b logical-and c overbar Equation m 6 1 0
1 1 1 Equation a logical-and b logical-and c Equation m 7 1 0

The minterms for which the function produces a Equation 1 are called 1-minterms. Likewise, minterms for which the function produces a Equation 0 are called 0-minterms. Any Boolean function can be expressed as sum of its 1-minterms. For example, consider the following function:

Equation f left-parenthesis a comma b comma c right-parenthesis equals a plus left-parenthesis b c right-parenthesis

The truth table for it is on the right. We can express this function as sum of 1-minterms:

Equation f left-parenthesis a comma b comma c right-parenthesis equals a overbar b c plus a b overbar c overbar plus a b overbar c plus a b c overbar plus a b c

We can replace the individual minterms with their respective index which is also shown in the table.

Equation f left-parenthesis a comma b comma c right-parenthesis equals m 3 plus m 4 plus m 5 plus m 6 plus m 7

This function can now be more concisely written a in the following way:

Equation f left-parenthesis a comma b comma c right-parenthesis equals sigma-summation m left-parenthesis list of 1 hyphen minterm indices right-parenthesis

I.e.

Equation f left-parenthesis a comma b comma c right-parenthesis equals m 3 plus m 4 plus m 5 plus m 6 plus m 7 equals sigma-summation m left-parenthesis 3 comma 4 comma 5 comma 6 comma 7 right-parenthesis

Likewise, the inverse of the function can be expressed as the sum of 0-minterms:

Equation StartLayout 1st Row 1st Column f prime left-parenthesis a comma b comma c right-parenthesis 2nd Column equals a overbar b overbar c overbar plus a overbar b overbar c plus a overbar b c overbar 2nd Row 1st Column Blank 2nd Column equals m 0 plus m 1 plus m 2 3rd Row 1st Column Blank 2nd Column equals sigma-summation m left-parenthesis 0 comma 1 comma 2 right-parenthesis EndLayout


Product of Sum (PoS)[edit]

Sum of Product Example
Function
Equation f left-parenthesis a comma b comma c right-parenthesis equals a logical-or left-parenthesis b logical-and c right-parenthesis
a b c minterms notation Equation f Equation f prime
0 0 0 Equation a logical-or b logical-or c Equation upper M 0 0 1
0 0 1 Equation a logical-or b logical-or c overbar Equation upper M 1 0 1
0 1 0 Equation a logical-or b overbar logical-or c Equation upper M 2 0 1
0 1 1 Equation a logical-or b overbar logical-or c overbar Equation upper M 3 1 0
1 0 0 Equation a overbar logical-or b logical-or c Equation upper M 4 1 0
1 0 1 Equation a overbar logical-or b logical-or c overbar Equation upper M 5 1 0
1 1 0 Equation a overbar logical-or b overbar logical-or c Equation upper M 6 1 0
1 1 1 Equation a overbar logical-or b overbar logical-or c overbar Equation upper M 7 1 0

A maxterm is the Boolean sum (ORing) of Equation n variables and contains all Equation n variables of the function just once, in either normal or complemented form. For example, for the function Equation f left-parenthesis a comma b right-parenthesis with two variables, we can have the following maxterms: Equation a logical-or b , Equation a logical-or b overbar , Equation a overbar logical-or b , and Equation a overbar logical-or b overbar .

A product term is the Boolean product (ANDing) of variables as a subset of the possible variables or their complements. For example, for function Equation f left-parenthesis a comma b comma c right-parenthesis , the following are possible product terms: Equation c overbar , Equation a logical-and b overbar , and Equation c logical-and a .

The sum of maxterms also called Product of Sum (PoS), canonical sum of products, maxterm expansion, canonical conjunctive normal form (CCNF) is a Boolean expression in which each term contains all the variables, either in normal or complemented form as the sum of all the maxterms.

Equation f left-parenthesis a comma b comma c comma d right-parenthesis equals a left-parenthesis b plus c plus d right-parenthesis

We can express that function in PoS form as follows.

Equation multiline equation line 1 f left-parenthesis a comma b comma c comma d right-parenthesis equals left-parenthesis a plus b plus c plus d right-parenthesis dot left-parenthesis a plus b plus c plus d overbar right-parenthesis dot left-parenthesis a plus b plus c overbar plus d right-parenthesis dot left-parenthesis a plus b plus c overbar plus d overbar right-parenthesis dot line 2 left-parenthesis a plus b overbar plus c plus d right-parenthesis dot left-parenthesis a plus b overbar plus c plus d overbar right-parenthesis dot left-parenthesis a plus b overbar plus c overbar plus d right-parenthesis dot left-parenthesis a plus b overbar plus c overbar plus d overbar right-parenthesis dot line 3 left-parenthesis a overbar plus b plus c plus d right-parenthesis

The maxterms for which the function produces a Equation 1 are called 1-maxterms. Likewise, maxterms for which the function produces a Equation 0 are called 0-maxterms. Any Boolean function can be expressed as product of its 0-maxterms. For example, consider the following function:

Equation f left-parenthesis a comma b comma c right-parenthesis equals a plus left-parenthesis b c right-parenthesis

The truth table for it is on the right. We can express this function as product of 0-maxterms:

Equation f left-parenthesis a comma b comma c right-parenthesis equals left-parenthesis a plus b plus c right-parenthesis dot left-parenthesis a plus b plus c overbar right-parenthesis dot left-parenthesis a plus b overbar plus c right-parenthesis

We can replace the individual minterms with their respective index which is also shown in the table.

Equation f left-parenthesis a comma b comma c right-parenthesis equals upper M 0 plus upper M 1 plus upper M 2

This function can now be more concisely written a in the following way:

Equation f left-parenthesis a comma b comma c right-parenthesis equals product upper M left-parenthesis list of 1 hyphen minterm indices right-parenthesis

I.e.

Equation f left-parenthesis a comma b comma c right-parenthesis equals upper M 0 plus upper M 1 plus upper M 2 equals product upper M left-parenthesis 0 comma 1 comma 2 right-parenthesis

Likewise, the inverse of the function can be expressed as the product of 0-maxterms:

Equation StartLayout 1st Row 1st Column f prime left-parenthesis a comma b comma c right-parenthesis 2nd Column equals left-parenthesis a plus b overbar plus c overbar right-parenthesis dot left-parenthesis a overbar plus b plus c right-parenthesis dot left-parenthesis a overbar plus b plus c overbar right-parenthesis dot left-parenthesis a overbar plus b overbar plus c right-parenthesis dot left-parenthesis a overbar plus b overbar plus c overbar right-parenthesis 2nd Row 1st Column Blank 2nd Column equals upper M 3 plus upper M 4 plus upper M 5 plus upper M 6 plus upper M 7 3rd Row 1st Column Blank 2nd Column equals product upper M left-parenthesis 3 comma 4 comma 5 comma 6 comma 7 right-parenthesis EndLayout


Minimization[edit]

Main article: logic minimization

Generating Boolean functions for certain logic is often pretty straightforward. On many occasions, however, those functions might be more complicated than they have to be. Minimization is the process of transforming a Boolean function into its smallest equivalent form. While the best circuit design depends on the technology involved, minimization usually translates directly into smaller circuits and lower implementation costs (areas, time, et al).

One way to minimize a Boolean expression is to simply massage the expression using the axioms and laws described earlier on. Such process is heuristic in nature and thus there is no one algorithm or rules that must be followed. For example, consider the following function Equation f left-parenthesis a comma b comma c comma d right-parenthesis equals a b plus b c plus b overbar c . It can be hand minimized as:

Equation StartLayout 1st Row 1st Column f left-parenthesis a comma b comma c comma d right-parenthesis 2nd Column equals a b plus b c plus b overbar c 2nd Row 1st Column Blank 2nd Column equals a b plus c left-parenthesis b plus b overbar right-parenthesis 3rd Column Blank 4th Column By Distributive Axiom 3rd Row 1st Column Blank 2nd Column equals a b plus c left-parenthesis 1 right-parenthesis 3rd Column Blank 4th Column By Inverse Axiom 4th Row 1st Column Blank 2nd Column equals a b plus c 3rd Column Blank 4th Column By Identity Axiom EndLayout

Karnaugh Map[edit]

Main article: Karnaugh Map

While minimization function through manual massaging of expressions works for basic functions, it becomes incredibly complex and time consuming as you increase the number of terms and variables involved. A Karnaugh Map (or K-Map) is technique that provides a systematic way to simplify as well as manipulate Boolean expressions.

Truth Table
Equation f left-parenthesis a comma b comma c right-parenthesis equals a b overbar plus c left-parenthesis a overbar plus b right-parenthesis
Inputs Outputs
A B C Q
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 1

A K-Map is a actually a truth table that has been rearranged in a number of important ways to make it possible to visually find the expression we're looking for. An Equation n -variable K-Map is table consisting of Equation 2 Superscript n cells, each representing a single minterm. For each minterm where the function results in a Equation 1 , we put a 1. Conversely, for each minterm where the function results in a Equation 0 , we put a zero - or more commonly, we leave blank.

3-var blank k-map.svg

Let's consider the following 3-variable Boolean function: Equation f left-parenthesis a comma b comma c right-parenthesis equals a b overbar plus c left-parenthesis a overbar plus b right-parenthesis . Because this function has 3 variables, we need a 3-variable K-Map which means we're working with an 8-cell table. Note the special arrangement of the columns and rows. Particularly, between two adjacent row and columns, only a single variable is allowed to transition from 0 to 1 or 1 to 0. As a result, after "01" we move to "11" instead of "10" which would result in both variables transitioning between 0 and 1. For 3 variables, the table is arranged in 4 columns of 2 rows each. 2 of the variables span the columns while the last variable spans the rows. Our choice for which variable goes where was arbitrary. Any way you want will work so long the k-map is constructed correctly based on a truth table.

a and not b or c k-map (1).svg

For every minterm in the truth table where the output is Equation 1 we mark 1 on the K-Map. The cells that result in Equation 0 are left blank for convenience. Once the k-map represents the entire truth table, visually inspect the table and locate the largest groups of adjacent squares containing 1. Those groups must have power of 2 number of cells; i.e. a group can only be of size 1, 2, 4, 8, etc.. A group of 3 cells thus must be broken down into 2 groups of 2 where one cell overlaps (see the k-map article for a more detailed explanation). In our K-Map, we have 1 group of 4 cells and another group of just 2 cells.

a and not b or c k-map (2).svg

For each group marked down we look which variables are common to all the cells in the group. For the group consisting of four cells neither a nor b are common since they both change, however c is always 1 for all four cells. Therefore the expression for that group is simply Equation c . For the second group with just two cells, c is no loner common to the group. However both a and b are now common since neither of them change. Because a is always 1 and b is always 0 in that group, the expression for the second group is Equation a b overbar . The final simplified expression is the SoP: Equation a b overbar plus c . I.e.:

Equation f left-parenthesis a comma b comma c right-parenthesis equals a b overbar plus c left-parenthesis a overbar plus b right-parenthesis equals a b overbar plus c

Quine-McCluskey Method[edit]

Main article: Quine-McCluskey Method

The Quine-McCluskey Method (QMM) is an algorithm developed for minimizing Boolean expressions. This algorithm is functionally identical to how a K-Map works but orients itself in tabular form. Due to its algorithmic nature, it's much more suitable to be implemented as a program and can be easily applied to any number of variables and terms. QMM is used in various EDA tools.

Boolean functions[edit]

Main article: Boolean Function

A Boolean function is an algebraic function of the form Equation f colon double-struck upper B Superscript n Baseline right-arrow double-struck upper B where Equation double-struck upper B equals StartSet 0 comma 1 EndSet is defined as Boolean domain and Equation n greater-than-or-equal-to 0 . There are Equation 2 Superscript 2 Super Superscript n possible Boolean functions for Equation n Boolean variables.

Properties of functions[edit]

Main article: Boolean Function Properties

The properties of Boolean functions have been a subject of extensive research especially in conjunction with the switching theory. Understanding the properties of the Boolean functions has proven to help in various stages of logic design (e.g. logic synthesis). Below are some of the more important properties of Boolean functions.

Duality Principle[edit]

Main article: Duality Principle

Earlier on it was pointed out that every axiom and every law has an OR form and an AND form. The Duality Principle simply state that when you take a valid Boolean statement and interchange all Equation logical-and with Equation logical-or and Equation 1 with Equation 0 and vice vesa, you obtain its dual which is also a valid Boolean statement.

Consider the following statement Equation ModifyingAbove left-parenthesis a logical-and b right-parenthesis With bar equals a overbar logical-or b overbar which happens to be DeMorgan's Law. Then by the duality principle we also know that Equation ModifyingAbove left-parenthesis a logical-or b right-parenthesis With bar equals a overbar logical-and b overbar must also be true. Indeed that is the second form of the law.

Self-dual function[edit]

Main article: Self-dual Boolean function

A Boolean function is said to be a self-dual function if it is equivalent to the same function with all inputs and outputs inverted. For example consider the majority function. It is defined as:

Equation MAJ left-parenthesis x comma y comma z right-parenthesis equals left-parenthesis x logical-and y right-parenthesis logical-or left-parenthesis x logical-and z right-parenthesis logical-or left-parenthesis y logical-and z right-parenthesis

We can derive it's dual function by inverting the inputs and outputs:

Equation StartLayout 1st Row 1st Column MAJ Superscript d Baseline left-parenthesis x comma y comma z right-parenthesis 2nd Column equals ModifyingAbove MAJ left-parenthesis x overbar comma y overbar comma z overbar right-parenthesis With bar 3rd Column By Duality Principle 2nd Row 1st Column Blank 2nd Column equals ModifyingAbove left-parenthesis x overbar logical-and y overbar right-parenthesis logical-or left-parenthesis x overbar logical-and z overbar right-parenthesis logical-or left-parenthesis y overbar logical-and z overbar right-parenthesis With bar 3rd Row 1st Column Blank 2nd Column equals ModifyingAbove left-parenthesis x overbar logical-and y overbar right-parenthesis With bar logical-and ModifyingAbove left-parenthesis x overbar logical-and z overbar right-parenthesis With bar logical-and ModifyingAbove left-parenthesis y overbar logical-and z overbar right-parenthesis With bar 3rd Column By DeMorgan prime s Law 4th Row 1st Column Blank 2nd Column equals left-parenthesis x logical-or y right-parenthesis logical-and left-parenthesis x logical-or z right-parenthesis logical-and left-parenthesis y logical-or z right-parenthesis 3rd Column By DeMorgan prime s Law 5th Row 1st Column Blank 2nd Column equals MAJ left-parenthesis x comma y comma z right-parenthesis EndLayout

Since Equation MAJ left-parenthesis x comma y comma z right-parenthesis equals MAJ Superscript d Baseline left-parenthesis x comma y comma z right-parenthesis , the majority function is said to be a self-dual function.

Monotonic functions[edit]

Main article: Monotonic Functions

The monotonicity property of Boolean functions says that for two Boolean vectors a and b such that Equation a Subscript n Baseline greater-than-or-equal-to b Subscript n Baseline for-all Subscript a comma b element-of StartSet 0 comma 1 EndSet Sub Superscript n ; if Equation f left-parenthesis a right-parenthesis greater-than-or-equal-to f left-parenthesis b right-parenthesis then Equation f is a monotonically increasing function. Conversely, if Equation f left-parenthesis a right-parenthesis less-than-or-equal-to f left-parenthesis b right-parenthesis then Equation f is a monotonically decreasing function. In other words, a monotonically increasing (or decreasing) function never has its output changes from one to a zero (or zero to a one for decreasing) by changing zero to a one in the input. Monotonic functions can be constructed with just ANDs and ORs operations without negation. By an extension, monotone circuits are circuits in which only AND and OR gates.

Unate functions[edit]

Main article: Unate Boolean Function
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Linear functions[edit]

Main article: Linear Boolean Functions

A linear function is either the constant Equation 0 function or the exclusive OR of Equation n variables. That is, is one that has the form

Equation f left-parenthesis x 0 comma x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis equals r 0 circled-plus circled-plus Underscript i equals 1 Overscript n Endscripts r Subscript i Baseline x Subscript i Baseline semicolon r Subscript i Baseline element-of double-struck upper B equals StartSet 0 comma 1 EndSet

For example, consider the following function Equation f left-parenthesis a comma b right-parenthesis equals a logical-and b logical-or a overbar logical-and b overbar . Since we can write it in the polynomial form above (i.e. Equation f equals 1 circled-plus a circled-plus b ), it's a linear function.

0/1 Preserving functions[edit]

Main article: 0-Preserving Boolean Function and 1-Preserving Boolean Function

A Boolean function is said to be 0-preserving if Equation f left-parenthesis 0 comma 0 comma ellipsis comma 0 right-parenthesis equals 0 . Likewise a Boolean function is said to be 1-preserving if Equation f left-parenthesis 1 comma 1 comma ellipsis comma 1 right-parenthesis equals 1 .

functional completeness[edit]

Main article: Functional Completeness

A set of Boolean functions Equation script upper F is said to be universal or complete iif it's 0-preserving, 1-preserving, Self-dual, monotonically increasing, and linear. A complete set can form any other Boolean expression by combining the functions in the set.

Complementary Function[edit]

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Additional Logical connectives[edit]

Main article: Logical connective

In additional to the basic AND, OR, and NOT, there are a number of other logical connectives.

XOR[edit]

Main article: xor
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conditional[edit]

Main article: conditional
Inputs Outputs
A B Q
0 0 1
0 1 1
1 0 0
1 1 1

It's often necessary to express things in the form Equation If upper A then upper B . In that form, Equation upper A right-arrow upper B is called a conditional, Equation upper A is the antecedent, and Equation upper B is the consequent. In order for the expression to hold, when Equation upper A is true, Equation upper B must also be true. However, conversely, when Equation upper A is false, the statement is trivially true regardless of Equation upper B .

biconditional[edit]

Main article: biconditional
Inputs Outputs
A B Q
0 0 1
0 1 0
1 0 0
1 1 1

A proposition in the form Equation upper A left-right-arrow upper B is called biconditional and represents the truth function: A if and only if B. The truth table is thus identical to Equation left-parenthesis upper A right-arrow upper B right-parenthesis logical-and left-parenthesis upper B right-arrow upper A right-parenthesis . In fact, the standard method of proving biconditional is to prove Equation upper A right-arrow upper B and then to prove Equation upper B right-arrow upper A .

Propositional calculus[edit]

Main article: propositional calculus
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Posets and Lattices[edit]

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Diagrammatic representations[edit]

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Venn Diagram[edit]

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Logic Diagram[edit]

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Hasse Diagram[edit]

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Additional topics[edit]

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See also[edit]


Facts about "Boolean Algebra"
instance ofbranch of algebra +
wikidata idQ173183 +