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Chapter 12: Problem 22

Using a logic table, verify each. $$(x y)^{\prime} \neq x^{\prime} y^{\prime}$$

Short Answer

Expert verified
Construct the truth table for the given expressions \((xy)'\) and \(x'y'\), then compare their outputs. The table shows: x | y | (xy)' | x'y' --|---|-------|----- T | T | F | F T | F | T | F F | T | T | F F | F | T | T Outputs for \((xy)'\) and \(x'y'\) are different for some input combinations, verifying the inequality \((xy)' \neq x'y'\).

Step by step solution

01

Construct truth table

Start by constructing a truth table with all possible input combinations for variables x and y. x | y --|-- T | T T | F F | T F | F
02

Calculate (xy)' output

Next, calculate the output for the given expression \((xy)'\). We will do this by performing the AND operation on x and y, and then applying the NOT operation on the result. x | y | xy | (xy)' --|---|----|------ T | T | T | F T | F | F | T F | T | F | T F | F | F | T
03

Calculate \(x'y'\) output

Now, calculate the output for the comparison expression \(x'y'\). To do this, perform the NOT operation on x and y separately, then perform the AND operation on the results. x | y | x' | y' | x'y' --|---|----|----|----- T | T | F | F | F T | F | F | T | F F | T | T | F | F F | F | T | T | T
04

Compare outputs and verify inequality

Finally, compare the outputs for the expressions \((xy)'\) and \(x'y'\) on the truth table. x | y | (xy)' | x'y' --|---|-------|----- T | T | F | F T | F | T | F F | T | T | F F | F | T | T We can see that the outputs for \((xy)'\) and \(x'y'\) are different for some of the input combinations, so it can be verified that \((xy)' \neq x'y'\), which is the desired inequality.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logic Tables
Logic tables, often called truth tables, are essential tools in Discrete Mathematics, particularly when dealing with logical expressions. They allow you to systematically list all possible combinations of truth values for logical variables. Logic tables make it easier to analyze logical statements and verify properties, such as inequalities, by showing the exact outcome of logical operations.

To construct a logic table, start with identifying the logical variables involved. Then, list all possible truth value combinations for these variables. Each row corresponds to one set of values for the variables, covering every possible scenario. This sets the foundation for analyzing how different logical operations affect these variables.
Truth Tables
Truth tables are a specific type of logic table used to determine truth values of complex logical expressions based on simpler components. Each row in a truth table corresponds to a unique combination of truth values for the given variables and reflects the outcome of the logical operation.

When creating a truth table, vary the input values systematically. Use columns to represent each variable. Add additional columns for each intermediate step of the logical operation, until you reach the final expression. This approach provides a clear view of how the input values affect the outcome, making it easier to draw conclusions about the logical expression.
Boolean Operations
Boolean operations are fundamental actions used in logic tables and truth tables to evaluate logical expressions. The primary operations include AND, OR, and NOT. Each operation has specific rules for how it affects truth values.
  • AND operation: Requires both operands to be true for the result to be true. Symbolically, this is expressed as \(x \land y\).
  • OR operation: Results in true if at least one operand is true. It is denoted as \(x \lor y\).
  • NOT operation: Inverts the truth value of a single operand so that true becomes false and false becomes true, represented by \(eg x\).
Boolean operations are combined in various ways to express more complex logical statements. Understanding how each fundamental operation works is crucial for analyzing logical expressions accurately in logic tables.
Logical Inequalities
Logical inequalities occur when the truth values of two logical expressions differ for at least one combination of input values. They help in analyzing when and why two expressions yield different outcomes. In the context of logic tables, verifying an inequality, like \((xy)' eq x'y'\), involves comparing the output columns of the two expressions.

To verify logical inequalities, construct truth tables for both expressions. Look for rows where the output values of the expressions differ, which indicates inequality. This ensures that the conclusions drawn about the logical relationships are accurate and valid, providing clarity and understanding of complex logical expressions.

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Most popular questions from this chapter

Using a Karnaugh map, simplify each boolean expression. $$x y^{\prime} z+x y^{\prime} z^{\prime}$$

Determine if each is a boolean expression, where each variable is boolean. $$ x^{\prime}+y z $$

Prove algebraically. $$(x+y) z=x z+y z$$

Find the boolean expression represented by each Karnaugh map.

Find the DNFs of the boolean functions in Exercises \(27-34\) $$ \begin{array}{|c|c|c|}\hline x & {y} & {f(x, y)} \\ \hline 0 & {0} & {1} \\\ {0} & {1} & {1} \\ {1} & {0} & {1} \\ {1} & {1} & {0} \\ \hline\end{array} $$
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