https://file.aiquickdraw.com/mathbot/file/1726167591727hufx6u79.jpg

Let's break down and solve the problem step by step.

### Step 1: Understanding the Propositions

We have the following compound propositions:

- \( A : (s \oplus t) \land (\neg s \lor u) \)
- \( B : (s \lor t) \oplus (t \land u) \)
- \( C : \neg (s \land u) \)

Here:
- \( \oplus \) denotes the **exclusive OR (XOR)** operation.
- \( \land \) is the logical **AND**.
- \( \lor \) is the logical **OR**.
- \( \neg \) is **NOT**.

The task requires:
1. **Constructing a truth table** for these three propositions.
2. **Evaluating the logical implications** between pairs of \( A \), \( B \), and \( C \), and determining if any of them are logically equivalent.

### Step 2: Constructing the Truth Table

The variables \( s \), \( t \), and \( u \) represent Boolean variables. For a truth table, we'll need to evaluate each proposition over all possible combinations of truth values for \( s \), \( t \), and \( u \) (i.e., 8 rows as each variable can be either True or False).

Let me compute this truth table.Here is the truth table for the propositions \( A \), \( B \), and \( C \):

| \( s \) | \( t \) | \( u \) | \( A \)   | \( B \)   | \( C \)   |
|--------|--------|--------|--------|--------|--------|
| False  | False  | False  | False  | False  | True   |
| False  | False  | True   | False  | False  | True   |
| False  | True   | False  | True   | True   | True   |
| False  | True   | True   | True   | False  | True   |
| True   | False  | False  | False  | True   | True   |
| True   | False  | True   | True   | True   | False  |
| True   | True   | False  | False  | True   | True   |
| True   | True   | True   | False  | False  | False  |

### Step 3: Analyzing the Logical Implications

We need to check the logical implications between pairs of \( A \), \( B \), and \( C \). The six implications are:

1. \( A \Rightarrow B \)
2. \( A \Rightarrow C \)
3. \( B \Rightarrow C \)
4. \( C \Rightarrow A \)
5. \( C \Rightarrow B \)
6. \( B \Rightarrow A \)

I'll now compute the truth values for each of these implications based on the truth table.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, if you would like to proceed manually:
- We already have the truth table for \( A \), \( B \), and \( C \).
- You can compute the implications by checking if \( A \Rightarrow B \), \( A \Rightarrow C \), and others hold for each row by verifying the condition \( \neg P \lor Q \).

If you'd like more clarification on how to do this or need help with anything else, feel free to ask!

### Here are some additional related questions:
1. How do you calculate the XOR of two propositions?
2. What is the general rule for calculating implications in logic?
3. Could \( A \) and \( B \) be logically equivalent, given their truth values?
4. How can we determine if two compound propositions are logically equivalent?
5. What role does negation play in logical implications?

**Tip:** When working with truth tables, always double-check that the basic operations (AND, OR, XOR, etc.) are applied correctly for each row of the table.

Let A, B, and C be the compound propositions A: (s ⊕ t) ∧ (¬s ∨ u), B: (s ∨ t) ⊕ (t ∧ u), C: ¬(s ∧ u). 1. Make a truth table for A, B, and C. 2. Analyze the logical implications between pairs of A, B, and C.

Explore propositional logic with a detailed truth table for compound propositions A: (s ⊕ t) ∧ (¬s ∨ u), B: (s ∨ t) ⊕ (t ∧ u), and C: ¬(s ∧ u). Analyze six logical implications between A, B, and C. Learn about logical equivalence and truth tables.

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