**Logical Expressions and De Morgan’s Laws**12. **Creating a Truth Table for De Morgan's Laws**In this exercise, we'll demonstrate that this version of De Morgan’s Laws is a tautology by constructing a truth table. The dual of the logical expression is provided below. Begin by drafting truth tables for both the original expression and its dual. Verification and explanation will reveal that these expressions are equivalent.### Original Expression:\[ a \lor b \land (b \lor \sim c) \]### Task:1. **Create truth tables for both expressions**: Construct a table for the original expression and another for its dual to show their equivalence.2. **Add Parentheses**: To clarify the precedence of the logic operators in computations, add parentheses as needed.3. **Verify equivalence**: Verify and explain how the comparison of truth tables shows equivalence between the original expression and its dual.**Additional Expression for Consideration:**\[ a \lor b \land \sim c \rightarrow \sim (a \lor b) \leftrightarrow a \land b \]### Task:1. **Insert Parentheses**: Add parentheses to indicate the proper order of operations according to logical precedence.**Notes:**- The logical operators used are as follows: - $ \lor $: OR - $ \land $: AND - $ \sim $: NOT - $ \rightarrow $: IMPLICATION - $ \leftrightarrow $: EQUIVALENCEEnsure calculations respect these precedences, especially while compiling truth tables.

To Show: De Morgan’s law is a tautology De Morgan’s law i ~p∧q≡ ~p∨~qii~p∨q≡ ~p∧~q We will draw…

12. Draw a truth table to show that this version of De Morgan's Laws is a tautology. You can use the Word table feature to lay out the table. Give the dual of the following logical expression: a v b^ (b V ~c) Create truth tables for both the original expression and the dual. Verify and explain how this illustrates that the original expression and the dual are equivalent. (Keep in mind that you may need to add parentheses to reflect the order in which the operations would be performed, according to the precedence of the logic operators.) Also, Consider the following expression with minimal parentheses a v bA nc → (a V b) → a ^ b Add parentheses to reflect the order in which the operations would be performed,

12. Draw a truth table to show that this version of De Morgan's Laws is a tautology. You can use the Word table feature to lay out the table. Give the dual of the following logical expression: a v b^ (b V ~c) Create truth tables for both the original expression and the dual. Verify and explain how this illustrates that the original expression and the dual are equivalent. (Keep in mind that you may need to add parentheses to reflect the order in which the operations would be performed, according to the precedence of the logic operators.) Also, Consider the following expression with minimal parentheses a v bA nc → (a V b) → a ^ b Add parentheses to reflect the order in which the operations would be performed,

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Logical Expressions and De Morgan’s Laws**

12. **Creating a Truth Table for De Morgan's Laws**

In this exercise, we'll demonstrate that this version of De Morgan’s Laws is a tautology by constructing a truth table. The dual of the logical expression is provided below. Begin by drafting truth tables for both the original expression and its dual. Verification and explanation will reveal that these expressions are equivalent.

### Original Expression:

\[ a \lor b \land (b \lor \sim c) \]

### Task:

1. **Create truth tables for both expressions**: Construct a table for the original expression and another for its dual to show their equivalence.

2. **Add Parentheses**: To clarify the precedence of the logic operators in computations, add parentheses as needed.

3. **Verify equivalence**: Verify and explain how the comparison of truth tables shows equivalence between the original expression and its dual.

**Additional Expression for Consideration:**

\[ a \lor b \land \sim c \rightarrow \sim (a \lor b) \leftrightarrow a \land b \]

### Task:

1. **Insert Parentheses**: Add parentheses to indicate the proper order of operations according to logical precedence.

**Notes:**

- The logical operators used are as follows:
- $ \lor $: OR
- $ \land $: AND
- $ \sim $: NOT
- $ \rightarrow $: IMPLICATION
- $ \leftrightarrow $: EQUIVALENCE

Ensure calculations respect these precedences, especially while compiling truth tables.

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