Boolean Algebra

In this tutorial we will learning about Minterm and Maxterm.

A boolean variable and its complement are called literals.

Example

Boolean variable A and its complement ~A are literals.

Minterm is a product of all the literals (with or without complement).

Example

if we have two boolean variables X and Y

then

X.(~Y)

is a minterm

we can express complement ~Y as Y’

so, the above minterm can be expressed as

XY’

So, if we have two variables then the minterm will consists of product of both the variables

We can also create Minterm from the given values of the variables.**If value is 0 then we take the complement of the variable.If value is 1 then we take the variable as is.**

Example

if X, Y and Z are three boolean variables

having value

X = 0

Y = 1

and

Z = 0

then,

Minterm = X’YZ’

Note! X and Z are 0 so their complement are taken, Y is 1 so it is taken as is.

Let us check another example

if X and Y are two boolean variables

having value

X = 1

Y = 0

then,

Minterm = XY’

If there are two variables X and Y then both of them will appear in the product when forming Minterm.

Following are the steps to get the shorthand notation for minterm.

- Write the term consisting of all the variables
- Replace all complement variables like ~X or X’ with 0
- Replace all non-complement variables like X or Y with 1
- Express the decimal equivalent of the binary formed in the above steps

The decimal number is then written as a subscript of letter **m** where, small m denote minterm.

Lets see some example.

1. write the term consisting of all the variables

XY’

2. replace all complement variables with 0

so, Y’ is replaced by 0.

3. replace all non-complement variables with 1

so, X is replaced by 1.

4. express the decimal equivalent of the binary formed in the above steps

XY’ = 10 in binary

(10)_{2} in decimal is 2.

so, shorthand notation of XY’ is m_{2}

Another example

1. write the term consisting of all the variables

AB’C

2. replace all complement variables with 0

So, B’ is replaced by 0.

3. replace all non-complement variables with 1

So, A and C are replaced by 1.

4. express the decimal equivalent of the binary formed in the above steps

AB’C = 101 in binary

(101)_{2} in decimal is 5.

so, shorthand notation of AB’C is m_{5}

Maxterm is a sum of all the literals (with or without complement).

Example

if we have two boolean variables X and Y

then

X + (~Y)

is a maxterm

we can express complement ~Y as Y’

so, the above maxterm can be expressed as

X + Y’

So, if we have two variables then the maxterm will consists of sum of both the variables.

We can also create Maxterm from the given values of the variables.**If value is 1 then we take the complement of the variable.If value is 0 then we take the variable as is.**

Example

if X, Y and Z are three boolean variables

having value

X = 0

Y = 1

and

Z = 0

then,

Maxterm = X + Y’ + Z

Note! Y is 1 so its complement is taken, X and Z are 0 so they are taken as is.

let us check another example

if X and Y are two boolean variables

having value

X = 1

Y = 0

then,

Maxterm = X’ + Y

If there are two variables X and Y then both of them will appear in the sum when forming maxterm.

Following are the steps to get the shorthand notation for maxterm.

- Write the term consisting of all the variables
- Replace all complement variables like ~X or X’ with 1
- Replace all non-complement variables like X or Y with 0
- Express the decimal equivalent of the binary formed in the above steps

The decimal number is then written as a subscript of letter **M** where, capital M denote maxterm.

1. write the term consisting of all the variables

X’+Y

2. replace all complement variables with 1

so, X’ is replaced by 1.

3. replace all non-complement variables with 0

so, Y is replaced by 0.

4. express the decimal equivalent of the binary formed in the above steps

X’+Y = 10 in binary

(10)_{2} in decimal is 2

so, shorthand notation of X’+Y is M_{2}

Lets check another example.

1. write the term consisting of all the variables

A’+B’+C

2. replace all complement variables with 1

so, A’ and B’ are replaced by 1

3. replace all non-complement variables with 0

so, C is replaced by 0

4. express the decimal equivalent of the binary formed in the above steps

A’+B’+C = 110 in binary

(110)_{2} in decimal is 6

so, shorthand notation of A’+B’+C is M_{6}

For **minterm**

replace all **complement** variables like ~X or X’ with **0**

replace all **non-complement** variables like X or Y with **1**

small letter **m** denote minterm.

For **maxterm**

replace all **complement** variables like ~X or X’ with **1**

replace all **non-complement** variables like X or Y with **0**

capital letter **M** denote maxterm.

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