

A. Construct the truth tables for the following compound statements.

1.p ^ (~q)

2. (p^q) ˅ [(~p) ˅q]



B.Determine whether each pair of statements is logically equivalent. Just write LOGICALLY EQUIVALENT or NOT. Show your solution/table.

### A. Constructing Truth Tables

#### 1. \( p \land (\neg q) \)
The compound statement \( p \land (\neg q) \) uses conjunction (\( \land \)) and negation (\( \neg \)). We'll build the truth table step by step.

| \( p \) | \( q \) | \( \neg q \) | \( p \land (\neg q) \) |
|--------|--------|-------------|----------------------|
|   T    |   T    |      F      |          F           |
|   T    |   F    |      T      |          T           |
|   F    |   T    |      F      |          F           |
|   F    |   F    |      T      |          F           |

#### 2. \( (p \land q) \lor [(\neg p) \lor q] \)
This compound statement involves conjunction (\( \land \)), disjunction (\( \lor \)), and negation (\( \neg \)). We'll also break this into parts.

| \( p \) | \( q \) | \( p \land q \) | \( \neg p \) | \( (\neg p) \lor q \) | \( (p \land q) \lor [(\neg p) \lor q] \) |
|--------|--------|----------------|-------------|------------------------|--------------------------------------|
|   T    |   T    |       T        |      F      |            T           |                 T                    |
|   T    |   F    |       F        |      F      |            F           |                 F                    |
|   F    |   T    |       F        |      T      |            T           |                 T                    |
|   F    |   F    |       F        |      T      |            T           |                 T                    |

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### B. Logical Equivalence
Now, let's determine whether each pair of statements is logically equivalent by analyzing their truth tables.

#### Pair 1: \( p \land (\neg q) \) and \( (p \land q) \lor [(\neg p) \lor q] \)
The truth values for both expressions are different for at least one row in the truth tables.

- \( p \land (\neg q) \): {F, T, F, F}
- \( (p \land q) \lor [(\neg p) \lor q] \): {T, F, T, T}

Therefore, the statements **are not logically equivalent**.

**Answer**: NOT LOGICALLY EQUIVALENT

Would you like to see the step-by-step breakdown for any part of this? Feel free to ask!

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### Follow-up Questions:
1. How does negation (\( \neg \)) affect logical statements?
2. What is the difference between conjunction (\( \land \)) and disjunction (\( \lor \)) in logical statements?
3. Can a statement ever be true when its conjunction includes a false term?
4. How do you verify logical equivalence using truth tables?
5. Are there any shortcuts to determine whether two logical statements are equivalent?

**Tip**: Conjunctions (\( \land \)) require both terms to be true for the entire expression to be true; disjunctions (\( \lor \)) require only one term to be true.

A. Construct the truth tables for the following compound statements. 1. p ^ (~q) 2. (p ^ q) ˅ [(~p) ˅ q] B. Determine whether each pair of statements is logically equivalent. Just write LOGICALLY EQUIVALENT or NOT. Show your solution/table.

Learn how to construct truth tables for compound logical statements like p ^ (~q) and (p ^ q) ˅ [(~p) ˅ q]. This detailed solution explains the steps for evaluating logical equivalence and provides insights into key concepts such as conjunction, disjunction, and negation. Suitable for students in Grades 9-12.

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