Problem 36 - Discrete Mathematics with Applications

Question

Find the DNF of each boolean function. $$ f(x, y)=x+x y^{\prime} $$

Step-by-Step Solution

Answer

The Disjunctive Normal Form of the given function is: $$ f(x, y) = (x' y')+(x y')+(x y) $$

Step 1: Identify the terms where the function is true

First, let's examine the given Boolean function: $$ f(x, y) = x + x' y $$ We can observe that the function is true for all cases when x is true, and also true when x is false and y is false. We can represent these cases in a truth table: ``` x | y | f(x, y) --------- 0 | 0 | 1 0 | 1 | 0 1 | 0 | 1 1 | 1 | 1 ``` From the truth table, we can see that the function is true for these cases: (x=0, y=0), (x=1, y=0), and (x=1, y=1).

Step 2: Construct minterms for each of these cases

Now, we'll construct minterms for each of the cases identified in Step 1. A minterm is a product of all input variables (inverted or not) such that the function is true for that case. Like this: 1. For (x=0, y=0): `x' y'` 2. For (x=1, y=0): `x y'` 3. For (x=1, y=1): `x y`

Step 3: Combine the minterms with OR operation

Finally, we combine all the minterms derived in Step 2 with OR operation to obtain the DNF: $$ DNF = (x' y')+(x y')+(x y) $$ So, the Disjunctive Normal Form of the given function is: $$ f(x, y) = (x' y')+(x y')+(x y) $$

Key Concepts

Truth Table

A truth table is a powerful tool used to represent all possible values of a boolean function. It lets you analyze how a function behaves under every combination of its input variables. For our function \( f(x, y) = x + x'y \), we have two variables, \( x \) and \( y \). This means there are four possible combinations of inputs, which translates to four rows in our truth table.

• When \( x = 0 \) and \( y = 0 \), the function results in 1. It shows that the function is true.
• For \( x = 0 \), \( y = 1 \), the function is false, represented by 0.
• When \( x = 1 \), \( y = 0 \), the output is 1, indicating truth.
• Both \( x = 1 \) and \( y = 1 \), again result in 1.

Each row in the truth table corresponds to a different combination of input variables and how our function responds to it. This is an essential first step in simplifying or finding the standard forms of Boolean expressions.

Disjunctive Normal Form

Disjunctive Normal Form (DNF) is a way to express a Boolean function as an OR of ANDs. Essentially, DNF consists of one or more OR terms, where each term is a conjunction (AND) of literals.In our exercise, the DNF of the function \( f(x, y) = x + x'y \) is derived by finding all the minterms that make the function true and combining them with the OR operation. In essence, each minterm corresponds to one "true" path in the truth table.To achieve DNF:

• Identify all the input combinations (from the truth table) that make the function evaluate to true.
• For each of these combinations, write a minterm representing the AND of literals. Each variable may appear as itself or its complement, depending on whether its corresponding input in the combination is 1 or 0.
• Combine all these minterms using the OR operator.

The result is a structured, clear expression that is particularly useful for constructing logic circuits or for further simplification.

Minterms

Minterms play a critical role when converting Boolean functions to their Disjunctive Normal Form. A minterm is an AND combination of all input variables, though each variable can appear in either its true or complemented form.In the Boolean function \( f(x, y) \), we identified the input combinations resulting in a true (1) value from the truth table. For each true output, a minterm is constructed as follows:

• When \( x = 0 \) and \( y = 0 \), the corresponding minterm is \( x'y' \).
• For \( x = 1 \) and \( y = 0 \), the minterm becomes \( xy' \).
• Lastly, when both \( x = 1 \) and \( y = 1 \), the minterm constructed is \( xy \).

Each minterm represents a specific combination of input variables that results in the function being true. By OR’ing all these minterms, you can construct the DNF of the function, which is a key step in Boolean function simplification and optimization.