Problem 29 - Discrete Mathematics with Applications

Question

Find the DNFs of the boolean functions $$\begin{array}{|ll||c|} \hline \boldsymbol{x} & \boldsymbol{y} & \boldsymbol{f}(\boldsymbol{x}, \boldsymbol{y}) \\ \hline 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 1 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The Disjunctive Normal Form (DNF) of the given boolean function \(f(x, y)\) is: $$f(x, y) = (\overline{x} \cdot \overline{y}) \vee (x \cdot y)$$

Step 1: Identify rows with output 1

In the given truth table, the output is 1 for the following rows: - \(x = 0\) and \(y = 0\) - \(x = 1\) and \(y = 1\)

Step 2: Create terms for each row with output 1

Now, we will create terms to represent these rows that had output 1 in the truth table: - For the row where \(x = 0\) and \(y = 0\), the term is \(\overline{x} \cdot \overline{y}\). - For the row where \(x = 1\) and \(y = 1\), the term is \(x \cdot y\).

Step 3: Combine the terms using OR operation

Finally, we will combine both terms using the OR operation to form the Disjunctive Normal Form (DNF) of the boolean function \(f(x, y)\): $$f(x, y) = (\overline{x} \cdot \overline{y}) \vee (x \cdot y)$$

Key Concepts

Disjunctive Normal Form (DNF)

Disjunctive Normal Form (DNF) is a way to express boolean functions using a standardized format. It's a type of logical formula where the expression is a disjunction (OR) of one or more conjunctions (AND) of literals. In simpler terms, a boolean function in DNF is basically a series of "AND" operations joined by "OR" operations.
This form is particularly useful because it simplifies logical evaluations and makes it easier to compare boolean functions. When creating a DNF expression, you look for instances when the function yields a true value (1 in logic circuits or tables).

• The conjunction for each true value is created by AND-ing the literals, where each input variable is in its true or complemented form, according to the condition which leads to true as output.
• Then, each of these conjunctions is joined by OR to form the complete DNF.

In the exercise, the DNF for the boolean function given in the truth table is \(f(x, y) = (\overline{x} \cdot \overline{y}) \vee (x \cdot y)\), capturing all instances where the function equals 1.

Truth Table

A truth table is a mathematical table used to determine the outcome of a boolean function by listing all possible values of its variables. Each row in the table represents a possible combination of input values and their corresponding output.
For a basic example of a truth table, consider a boolean function with two variables, x and y:

• The input part of the table displays all combinations of x and y, typically using binary values: 0 (false) and 1 (true).
• The output part of the table shows the function's result based on these inputs.

This tool is valuable for easily understanding how different inputs affect the output of a boolean function. In the provided exercise, the truth table visually represents the function \(f(x, y)\) for variables with four possible combinations of x and y, clearly identifying where the function outputs a 1.
The table is structured as follows: for \(x = 0, y = 0\) and \(x = 1, y = 1\), the output is 1, which is crucial for determining the DNF of the function.

Boolean Algebra

Boolean Algebra is a branch of mathematics dealing with variables that have two possible values: true and false, often represented as 1 and 0. This form of algebra uses operations like AND, OR, and NOT, which emulate logical processes in computer circuits and other logic-based systems.
In Boolean Algebra:

• An AND operation is represented by \(\cdot\) and returns true only if both operands are true.
• An OR operation, denoted by \(\vee\), results in true if any operand is true.
• The NOT operation is shown with an overline (\(\overline{x}\)), and inverts the value of the operand, turning true into false, and false into true.

These operations are foundational in constructing boolean expressions like DNFs. The key to mastering Boolean Algebra is understanding the basic identities and laws, such as De Morgan's Theorems, which can simplify expressions and foster efficient boolean function evaluations.
In the given exercise, Boolean Algebra is used to convert data from a truth table into a DNF expression by applying these operations to identify and express the true outputs.