Question

Given \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{yz}(\mathrm{x}+\mathrm{y})+(\mathrm{z}+\mathrm{x})(\mathrm{y}+\mathrm{z})\), put \(\mathrm{f}\) into (1) the disjunctive normal form \((\mathrm{dnf})\) (2) the full disjunctive normal form (3) the sum-of-products form (4) the expanded sum-of-products.

Short answer

Rewriting the given function in different forms, we have: (1) Disjunctive Normal Form (DNF): \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z}) = 2\mathrm{xyz} + 2\mathrm{y^2z} + \mathrm{x^2z} + \mathrm{xy^2} + \mathrm{xz^2}\) (2) Full Disjunctive Normal Form cannot be obtained for the given function. (3) Sum-of-Products Form: \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z}) = 2\mathrm{xyz} + 2\mathrm{y^2z} + \mathrm{x^2z} + \mathrm{xy^2} + \mathrm{xz^2}\) (4) Expanded Sum-of-Products: \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z}) = 2\mathrm{xyz} + 2\mathrm{y^2z} + \mathrm{x^2z} + \mathrm{xy^2} + \mathrm{xz^2}\)

Step-by-step solution

Rewrite the given function

We are given the function \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{yz}(\mathrm{x}+\mathrm{y})+(\mathrm{z}+\mathrm{x})(\mathrm{y}+\mathrm{z})\). Let's distribute and then simplify. \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z}) = \mathrm{xyz} + \mathrm{y^2z} + \mathrm{xyz} + \mathrm{x^2z} + \mathrm{xy^2} + \mathrm{xz^2} + \mathrm{y^2z}\) Now combine like terms: \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z}) = 2\mathrm{xyz} + 2\mathrm{y^2z} + \mathrm{x^2z} + \mathrm{xy^2} + \mathrm{xz^2}\)

(1) Disjunctive Normal Form (DNF)

In this step, we want to convert the function \(\mathrm{f}\) into disjunctive normal form (DNF). A function is in DNF if it is a disjunction (OR) of conjunctions (AND) of literals (a variable or its negation). In this case, the function is already in DNF since it is a sum of products of variables. The DNF for the given function is: \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z}) = 2\mathrm{xyz} + 2\mathrm{y^2z} + \mathrm{x^2z} + \mathrm{xy^2} + \mathrm{xz^2}\)

(2) Full Disjunctive Normal Form

The full disjunctive normal form is a specific type of DNF where every term contains each variable exactly once, either in its normal form or in its negation (complement). We cannot rewrite the given function in full DNF form because some variables are squared. Therefore, the function cannot be put into full disjunctive normal form as it's given.

(3) Sum-of-Products Form

The sum-of-products form is like the DNF, but also allows constants that can be factored out of each product term, and is already in that form: \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z}) = 2\mathrm{xyz} + 2\mathrm{y^2z} + \mathrm{x^2z} + \mathrm{xy^2} + \mathrm{xz^2}\)

(4) Expanded Sum-of-Products

Finally, the expanded sum-of-products form is a version of the sum-of-products form where each product term is multiplied by a unique constant to create distinct product terms. In this case, the given function actually has its product terms already multiplied by constants, so it is also its own expanded sum-of-products form: \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z}) = 2\mathrm{xyz} + 2\mathrm{y^2z} + \mathrm{x^2z} + \mathrm{xy^2} + \mathrm{xz^2}\)

Key concepts

sum-of-products form

The sum-of-products form is an important representation in Boolean algebra that helps to simplify logical expressions. This form involves expressing a complex Boolean function as a series of AND operations, known as products, which are then summed using OR operations. Each product is a conjunction of literals, which can be variables or their complements.

This expression is particularly useful because it allows for easy simplification and is often used as an intermediary step in deriving a desired logical form or evaluating logical circuits.

• It comprises sum terms using the OR operator.
• Each sum term is a product of literals.
• The form helps in systematic simplification of logic expressions.

In the original exercise, we can see that the given function is written directly in a sum-of-products form: \(2xyz + 2y^2z + x^2z + xy^2 + xz^2\). Each term here represents a product, and they are all summed up to form the function.

Boolean algebra

Boolean algebra serves as the foundation for digital circuits and logic design. It is a branch of algebra that deals with variables having two distinct values, typically referred to as true or false (or 0 and 1 in binary terms). Boolean algebra uses logical operations such as AND, OR, NOT, NAND, NOR, XOR, and XNOR to manipulate these binary values. The key aspects of Boolean algebra include:

• Variables are binary (either 0 or 1).
• Operations follow specific rules, such as negation (NOT), conjunction (AND), and disjunction (OR).
• Boolean expressions can be simplified using identities like De Morgan's Theorems.

By applying Boolean algebra, we can transform complex logic expressions into simpler forms, like the sum-of-products form. This aids in analyzing and designing efficient logic circuits. In the context of the original exercise, using Boolean algebra allows us to express the function \(f(x, y, z)\) in various forms that are useful for implementation or further analysis.

logic expressions

Logic expressions are mathematical representations that describe logical operations and relationships between Boolean variables. These expressions are crucial in computer science, electrical engineering, and mathematics for modeling decision-making processes. For example, logic expressions appear in digital circuits for tasks like arithmetic operations, data processing, and control systems.

• Logic expressions utilize Boolean variables and operations like AND, OR, NOT.
• They are often used to represent real-world decision-making criteria.
• Such expressions can be converted across different forms, such as canonical forms, to ease manipulation.

The expression \(f(x, y, z)\) provided in the original problem exemplifies a complex logic expression that undergoes transformations to achieve a desired state like the sum-of-products form. Understanding these transformations is key to mastering logical operations and utilizing them in practical applications such as automated reasoning and digital circuit design.

expanded sum-of-products

The expanded sum-of-products form is a specific representation where each product term in a sum-of-products form is multiplied by a distinct constant. This form ensures that each term in the expression is unique in terms of its coefficient, which adds specificity to the logical expression.

• Each term has a unique coefficient.
• This form preserves distinctions between literal combinations.
• It's especially useful in detailed logical analysis and optimizations.

In cases such as the exercise presented, the expression \(2xyz + 2y^2z + x^2z + xy^2 + xz^2\) is already in its expanded form. This is because each product term involves coefficients that differentiate them, allowing for individual attention during the simplification or implementation phase in logic design. Recognizing the expanded sum-of-products form is helpful for professionals dealing with complex logical circuits or algebraic manipulations.