Boolean Functions - Lecture 5

A minterm is a product of literals representing a single combination of input variables, where each variable or its complement appears exactly once in the expression. Minterms are denoted by 'm' followed by a subscript number corresponding to the decimal equivalent of the binary representation of the input combination.

A maxterm is a sum of literals representing a single combination of input variables, where each variable or its complement appears exactly once in the expression. Maxterms are denoted by 'M' followed by a subscript number corresponding to the decimal equivalent of the binary representation of the input combination.

DeMorgan's theorem states that the complement of a product is equal to the sum of the complements, and the complement of a sum is equal to the product of the complements. Formally: (x.y)’= x’+y’ and (x +y)’ = x’.y’

The minterm and maxterm for a given input combination are complements of each other. This relationship is a direct consequence of DeMorgan's theorem, which holds for any number of variables. For example, the complement of minterm 'm2' is maxterm 'M2', and vice-versa.

The index used to identify minterms and maxterms is directly derived from the binary representation of the corresponding input combination. The decimal equivalent of the binary pattern associated with a specific input determines the index.

A minterm is a product term that represents one specific combination of input values. It has a '1' value for that specific combination and '0' for all others.

A maxterm is a sum term that represents one specific combination of input values. It has a '0' value for that specific combination and '1' for all others.

In a minterm function table, each column corresponds to a specific input combination, and only one entry is a '1' for that specific combination.

In a maxterm function table, each column corresponds to a specific input combination, and only one entry is a '0' for that specific combination.

Representing a Boolean function as a sum of minterms, where each minterm corresponds to a '1' entry in the function table.

Representing a Boolean function as a product of maxterms, where each maxterm corresponds to a '0' entry in the function table.

The standard form of a Boolean function expressed as a sum of minterms.

A Boolean function expressed as a sum of minterms, where each minterm is represented by its index (m) in the minterm function table.

The index of a minterm or maxterm is its decimal representation, obtained from converting the binary pattern representing the term.

In a minterm, a '1' in the binary representation of the index indicates the corresponding variable is uncomplemented, while a '0' indicates it's complemented.

In a maxterm, a '0' in the binary representation of the index indicates the corresponding variable is uncomplemented, while a '1' indicates it's complemented.

Minterms and maxterms are always represented with all variables present, and the variables are listed in the same order, typically alphabetically.

A term that doesn't include all variables is not a minterm, and a term that's not a sum of all variables is not a maxterm.

The minterm and maxterm representations of a Boolean expression provide a systematic way to analyze and express the truth values of the expression.

The minterms that correspond to the function having a value of '1'.

The maxterms that correspond to the function having a value of '0'.

Converting between SOM and POM involves finding all the missing minterms or maxterms, respectively, and using their indices to create the complementary expression.

Expressed as an OR of AND terms.

Expressed as an AND of OR terms.

A representation that is neither purely SOP nor POS, often containing mixed AND and OR operations.

A logic circuit where AND gates are followed by a single OR gate. It's used to represent a sum-of-products expression.

A logic circuit where OR gates are followed by a single AND gate. It's used to represent a product-of-sums expression.

A logic circuit implementation where the logic function has a single AND gate followed by a single OR gate, with no additional gates in between. All input variables are in their original or inverted forms.

A logic function where all terms are expressed as minterms or maxterms in a standard format. They are also called Canonical forms.

A logic circuit implementation where a function has at least three levels of gates. It typically involves more gates than a two-level implementation.

A logic circuit where the output is determined by the inputs based on the truth table.

The simplest form of implementing standard forms, where the output of the first level of gates is used as input for the second level. No intermediate level of gates or any additional logic exists between the first level of gates and the second level of gates.

A sum-of-minterms (SOM) expression is a canonical form of a Boolean function where the function is represented as the sum of minterms. Each minterm corresponds to a row in the truth table where the output is '1'.

A product-of-maxterms (POM) expression is a canonical form of a Boolean function where the function is represented as the product of maxterms. Each maxterm corresponds to a row in the truth table where the output is '0'.

A function in SOP (Sum-of-Products) form can be converted to POS (Product-of-Sums) form by considering the complements of minterms. The minterms missing in the original SOM expression represent the maxterms in the equivalent POM expression.

The complement of a Boolean function is obtained by changing all the '1's to '0's and all the '0's to '1's in its truth table. This corresponds to selecting the minterms (or maxterms) not included in the original function's expression.

The shorthand notation for a function in SOM form is represented as ∑m(i, j, k...), where 'i', 'j', 'k'... are the indices of the minterms included in the expression. These indices correspond to the decimal representation of the binary values of input combinations in the truth table.

The shorthand notation for a function in POM form is represented as ∏M(i, j, k...), where 'i', 'j', 'k'... are the indices of the maxterms included in the expression. These indices correspond to the decimal representation of the binary values of input combinations in the truth table.