Problem 15 - Discrete Mathematics with Applications

Question

Find the DNF of each boolean function. $$f(x, y, z)=x \uparrow(y \uparrow z)$$

Step-by-Step Solution

Answer

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

Step 1: Evaluate the function using the truth table

For this boolean function, we have three input variables (x, y, z). We will list all possible combinations of input values and find the output of the function. | x | y | z | x NOR (y NOR z) | |---|---|---|-----------------| | 0 | 0 | 0 | | | 0 | 0 | 1 | | | 0 | 1 | 0 | | | 0 | 1 | 1 | | | 1 | 0 | 0 | | | 1 | 0 | 1 | | | 1 | 1 | 0 | | | 1 | 1 | 1 | | Now, let's evaluate the function for each row: \(f(x, y, z)=x \uparrow(y \uparrow z)\) |x | y | z | y NOR z | x NOR (y NOR z)| |---|---|---|---------|-----------------| | 0 | 0 | 0 | 1 | 0 | | 0 | 0 | 1 | 0 | 1 | | 0 | 1 | 0 | 0 | 1 | | 0 | 1 | 1 | 0 | 1 | | 1 | 0 | 0 | 1 | 0 | | 1 | 0 | 1 | 0 | 1 | | 1 | 1 | 0 | 0 | 1 | | 1 | 1 | 1 | 0 | 1 |

Step 2: Identify the TRUE rows

From the truth table, we can see that the function is TRUE when the input variables are: - (0, 0, 1) - (0, 1, 0) - (0, 1, 1) - (1, 0, 1) - (1, 1, 0) - (1, 1, 1)

Step 3: Create terms using input variables and their complements

For each TRUE row, we will create a term using the input variables and their complements. The complement of a variable is denoted by placing a bar over the variable, for example, \(\overline{x}\). Using AND operation (or conjunction), the terms are: 1. \(\overline{x}\overline{y}z\) 2. \(\overline{x}y\overline{z}\) 3. \(\overline{x}yz\) 4. \(x\overline{y}z\) 5. \(xy\overline{z}\) 6. \(xyz\)

Step 4: Combine the terms using OR operation

To form the DNF, we will now combine the terms using the OR operation (or disjunction): $$f(x, y, z) = \overline{x}\overline{y}z \vee \overline{x}y\overline{z} \vee \overline{x}yz \vee x\overline{y}z \vee xy\overline{z} \vee xyz$$ So, the DNF of the given boolean function is: $$f(x, y, z) = \overline{x}\overline{y}z \vee \overline{x}y\overline{z} \vee \overline{x}yz \vee x\overline{y}z \vee xy\overline{z} \vee xyz$$

Key Concepts

Boolean Function

A Boolean function is a mathematical expression that involves binary variables, each taking the value of either 0 or 1. These variables are combined using logical operations such as AND (conjunction), OR (disjunction), and NOT (negation), which correspond to the logical operations of multiplication, addition, and complement respectively.

For instance, in the exercise provided, the Boolean function is defined as:
\[\begin{equation}f(x, y, z)=x \uparrow (y \uparrow z)\end{equation}\], where \(\uparrow\) represents the NOR operation, a compound logical operation that behaves as the negation of OR.

In other words, the output of a Boolean function is also binary, being 1 if the expression holds true and 0 if it does not, given a particular set of input values.

Truth Table

A truth table is a tabular representation of all possible combinations of binary input values to a logical statement or a Boolean function, along with the corresponding output values. It exhaustively lists all possible scenarios that the input variables could encounter, and details the outcome for each scenario based on the defined logic or function.

Through the truth table, students can systematically determine the output of a Boolean function for each combination of input values. This visual tool is particularly essential in understanding the behavior of Boolean expressions and is a crucial stepping stone for formulating the Disjunctive Normal Form (DNF) of a given function, as seen in the exercise's solution.

Logical Operations

Logical operations are the fundamental building blocks of Boolean functions and include operations like AND, OR, NOT, NOR, NAND, XOR, and XNOR.

• AND (\(\land\)) - Gives a value of 1 if and only if all the operands are 1.
• OR (\(\lor\)) - Gives a value of 1 if at least one of the operands is 1.
• NOT (\(eg\)) - Inverts the value of the operand: 1 becomes 0, and 0 becomes 1.
• NOR - A combination of OR and NOT and gives a result that is the negation of OR.
• NAND - A combination of AND and NOT and gives a result that is the negation of AND.
• XOR - Gives 1 if and only if the operands differ.
• XNOR - The negation of XOR, gives 1 if the operands are the same.

In the exercise, the NOR logical operation is used within the provided Boolean function. By understanding these operations, students can compose and interpret complex Boolean expressions.