Problem 15 - Discrete Mathematics with Applications

Question

Construct a logic table for each boolean function defined by each boolean expression. $$x y z+x(y z)^{\prime}$$

Step-by-Step Solution

Answer

The truth table for the given Boolean expression \(x( yz+x(y z)^{\prime})\) is as follows: x | y | z | \(x( yz+x(y z)^{\prime})\) --|--|--|------------------------ 0 | 0 | 0 | 0 0 | 0 | 1 | 0 0 | 1 | 0 | 0 0 | 1 | 1 | 0 1 | 0 | 0 | 1 1 | 0 | 1 | 1 1 | 1 | 0 | 1 1 | 1 | 1 | 1

Step 1: List all possible combinations of x, y, and z.

Here are all 8 combinations of truth values for x, y, and z: 1. x = 0, y = 0, z = 0 2. x = 0, y = 0, z = 1 3. x = 0, y = 1, z = 0 4. x = 0, y = 1, z = 1 5. x = 1, y = 0, z = 0 6. x = 1, y = 0, z = 1 7. x = 1, y = 1, z = 0 8. x = 1, y = 1, z = 1

Step 2: Evaluate the expression for each combination.

Now, we will evaluate the given expression x( yz+x(y z)^{\prime}) for each combination of x, y, and z: 1. x = 0, y = 0, z = 0: 0(0*0+0(0*0')=0(0)=0 2. x = 0, y = 0, z = 1: 0(0*1+0(0*1')=0(0)=0 3. x = 0, y = 1, z = 0: 0(1*0+0(1*0')=0(1)=0 4. x = 0, y = 1, z = 1: 0(1*1+0(1*1')=0(1)=0 5. x = 1, y = 0, z = 0: 1(0*0+1(0*0')=1(1)=1 6. x = 1, y = 0, z = 1: 1(0*1+1(0*1')=1(1)=1 7. x = 1, y = 1, z = 0: 1(1*0+1(1*0')=1(1)=1 8. x = 1, y = 1, z = 1: 1(1*1+1(1*1')=1(2)=1

Step 3: Construct the truth table.

Finally, we can create the truth table by listing the input combinations and their corresponding output. The truth table for the given Boolean expression is as follows: x | y | z | x( yz+x(y z)^{\prime}) --|--|--|------------------------ 0 | 0 | 0 | 0 0 | 0 | 1 | 0 0 | 1 | 0 | 0 0 | 1 | 1 | 0 1 | 0 | 0 | 1 1 | 0 | 1 | 1 1 | 1 | 0 | 1 1 | 1 | 1 | 1

Key Concepts

truth table

A truth table is a mathematical table used to determine the output of a logical operation. It lists all possible combinations of inputs, and, for each combination, it shows the corresponding output. Truth tables are commonly used in Boolean logic to display how different combinations of truth values affect the outcome of a particular function.

To create a truth table for a Boolean expression, follow these steps:

• Identify all the variables in the expression. In our example, these are \(x\), \(y\), and \(z\).
• Calculate the total number of combinations. With three variables, there are \(2^3 = 8\) possible combinations.
• For each combination, evaluate the Boolean expression to find the resulting output.

This process allows us to visually represent the logic of the expression, making it easier to understand and analyze.

Boolean expression

A Boolean expression is a logical statement that can be evaluated to one of two values: true or false. In Boolean logic, these values are often represented by \(1\) and \(0\), respectively. Boolean expressions are constructed using Boolean variables, logical operators, and parentheses to indicate the desired order of operations.

• Common logical operators include AND (\( \cdot \)), OR (\(+\)), and NOT (\( \prime \)).
• The expression \(x(yz + x(yz)^{\prime})\) involves several operations: an AND operation within the parentheses, a NOT operation, and an OR operation.

To analyze a Boolean expression, you must evaluate it for all possible combinations of variable values, which results in a complete set of scenarios covered in a truth table. Each calculation, according to the operator rules, ultimately helps in predicting the condition in which the entire expression would be true or false.

logic evaluation

Logic evaluation is the process through which we determine the output of a Boolean expression by systematically assessing the influence of various inputs. Translating a Boolean expression into a truth table involves logic evaluation as a critical step.

Steps in logic evaluation include:

• Substitute each combination of input values into the Boolean expression.
• Apply the rules of Boolean algebra to simplify or calculate each expression.
• Conclude by determining the resulting output value (true or false) for each substitution.

For example, if we take the expression \(x(yz + x(yz)^{\prime})\), logic evaluation entails determining what the expression evaluates to when each of \(x\), \(y\), and \(z\) are given values, such as \(0\) or \(1\). By doing so, we create a comprehensive understanding of the expression's behavior, illustrating its function across all potential scenarios.