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Let's tackle these two problems step by step. We will construct the truth tables for both compound statements as required.

### 1. Construct a truth table for the compound statement \(\neg(p \land q) \leftrightarrow (\neg p \lor \neg q)\).

This is a biconditional statement. The first part is \(\neg(p \land q)\), and the second part is \(\neg p \lor \neg q\). The table will compare these two parts and check if they are logically equivalent (i.e., whether the entire expression is true or false).

#### Truth Table Construction:

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

**Conclusion:** Since the final column has all true values, this compound statement is a **tautology**.

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### 2. Construct a truth table for the compound statement \((p \land \neg q) \rightarrow (p \rightarrow q)\).

Here, we first need to compute \(p \land \neg q\) and \(p \rightarrow q\) before considering the implication \((p \land \neg q) \rightarrow (p \rightarrow q)\).

#### Truth Table Construction:

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

**Conclusion:** Since the final column has a mix of true and false values, this compound statement is a **contingency**.

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Let me know if you'd like further clarification or details on these steps. Here are some additional questions you can explore:

1. How would the truth table change if we modified \(\lor\) to \(\land\) in the first statement?
2. What logical operation can you infer from \(\leftrightarrow\)?
3. How would you define a contradiction based on the truth table outcomes?
4. Why is the second statement not a tautology or contradiction?
5. What role do negations play in these logical expressions?

**Tip:** When constructing truth tables, always calculate the intermediate steps carefully to ensure accurate results.

Construct truth tables for compound statements and determine if they are tautology, contradiction, or contingency.

Learn how to construct truth tables for logical statements including negation, conjunction, disjunction, and biconditional operators. Determine whether the compound statements are tautologies, contradictions, or contingencies. Suitable for students in Grades 10-12.

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