Find the truth table of the following statements.

(a) p vee( sim q ^ r )

(b) ( neg p ^ q ) p

p q

(d) sim p vee q

Let's tackle each statement and construct their corresponding truth tables.

We'll need to define a truth table for each of the variables \( p \), \( q \), and \( r \), since the statements involve combinations of these propositions. First, let's identify each logical symbol:
- \( \vee \): logical OR
- \( \wedge \): logical AND
- \( \neg \) or \( \sim \): logical NOT
- \( \Rightarrow \): logical implication (not used directly here but often present in logical exercises)

We’ll go through each expression one by one.

### (a) \( p \vee ( \sim q \wedge r ) \)
This is the logical OR between \( p \) and the conjunction of \( \sim q \) (the negation of \( q \)) and \( r \). Let’s construct the truth table:

| \( p \) | \( q \) | \( r \) | \( \sim q \) | \( \sim q \wedge r \) | \( p \vee ( \sim q \wedge r ) \) |
|:------:|:------:|:------:|:-----------:|:----------------------:|:------------------------------:|
|   T    |   T    |   T    |      F      |           F            |              T                 |
|   T    |   T    |   F    |      F      |           F            |              T                 |
|   T    |   F    |   T    |      T      |           T            |              T                 |
|   T    |   F    |   F    |      T      |           F            |              T                 |
|   F    |   T    |   T    |      F      |           F            |              F                 |
|   F    |   T    |   F    |      F      |           F            |              F                 |
|   F    |   F    |   T    |      T      |           T            |              T                 |
|   F    |   F    |   F    |      T      |           F            |              F                 |

### (b) \( ( \neg p \wedge q ) \)
This statement involves the negation of \( p \) combined with \( q \) using a logical AND.

| \( p \) | \( q \) | \( \neg p \) | \( \neg p \wedge q \) |
|:------:|:------:|:------------:|:---------------------:|
|   T    |   T    |      F       |           F           |
|   T    |   F    |      F       |           F           |
|   F    |   T    |      T       |           T           |
|   F    |   F    |      T       |           F           |

### (c) \( p \Rightarrow q \) (This seems to be a logical implication, which is not part of your question but might be inferred based on what you wrote. If not, clarify what "p p" means).

### (d) \( \sim p \vee q \)
This is the logical OR between the negation of \( p \) and \( q \).

| \( p \) | \( q \) | \( \sim p \) | \( \sim p \vee q \) |
|:------:|:------:|:------------:|:-------------------:|
|   T    |   T    |      F       |          T          |
|   T    |   F    |      F       |          F          |
|   F    |   T    |      T       |          T          |
|   F    |   F    |      T       |          T          |

---

This provides the full truth tables for each of the logical statements you asked for. 

Would you like a more detailed explanation of any of these tables? Here are some follow-up questions that might interest you:

1. How do we interpret truth values in complex logical statements?
2. What is the difference between \( \vee \) (OR) and \( \wedge \) (AND)?
3. How does the truth table of \( p \Rightarrow q \) look compared to other logical operators?
4. Can truth tables help in proving logical equivalences?
5. How can truth tables be used to check for tautologies or contradictions?

**Tip:** In a truth table, the number of rows is determined by \( 2^n \), where \( n \) is the number of distinct propositions (e.g., \( p \), \( q \), \( r \)).

Find the truth table of the following statements.
(a) p ∨ (¬q ∧ r)
(b) (¬p ∧ q)
(c) p ⇒ q
(d) ¬p ∨ q

Discover how to construct truth tables for logical statements involving OR (∨), AND (∧), NOT (¬), and implication (⇒). This step-by-step guide explores key concepts in propositional logic and provides complete truth tables for four different logical expressions. Ideal for undergraduate students studying logic or discrete mathematics.

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