This assignment focuses on Discrete Mathematics, specifically propositional logic, truth tables, and logical inference. Below is a breakdown of the tasks and what each one asks for:

1. Translating Propositions into Simple Sentences

Given:

• $p$: Adit is a model student.
• $q$: Dita is an outstanding student.

Interpret the following logical propositions into simple sentences:

• (a) $p \rightarrow q$: If Adit is a model student, then Dita is an outstanding student.
• (b) $\neg q \land p$: Dita is not an outstanding student, but Adit is a model student.
• (c) $\neg (p \leftrightarrow q)$: It is not true that Adit being a model student is equivalent to Dita being outstanding.

2. Inverse, Converse, and Contrapositive of a Conditional Statement

Given:

• "If the product of two numbers is odd, then both numbers are odd."

Let's rewrite the inverse, converse, and contrapositive of this statement.

• Inverse: If the product of two numbers is not odd, then at least one of them is not odd.
• Converse: If both numbers are odd, then their product is odd.
• Contrapositive: If at least one of the two numbers is not odd, then their product is not odd.

3. Completing the Truth Table with Custom Operator Δ

You need to fill in the truth table based on the definition of Δ. The following operations are given:

• $\sim q \Delta p$
• $p \Delta q$
• $\sim p \Delta q$
• $\sim (p \rightarrow q)$

This will involve systematically substituting values for $p$ and $q$ (True or False) and applying the custom operator Δ as defined.

4. Using a Truth Table to Prove a Tautology

You are asked to verify that:

• (a) $\left[(p \rightarrow q) \land (\neg q \rightarrow \neg r)\right] \rightarrow (p \rightarrow r)$ is a tautology.

  • Construct the truth table, and show that the statement is always true regardless of the values for $p$, $q$, and $r$.

• (b) $p \rightarrow (q \lor \neg r) \equiv (\neg p \lor q) \lor \neg r$.

  • Show that both expressions yield the same truth values across all possible cases of $p, q,$ and $r$.

• (c) $(p \lor \neg p) \equiv [(\neg p \land q) \rightarrow (p \leftrightarrow q)]$.

  • Prove the equivalence of both statements through the truth table.

5. Validating an Argument Using Inference Rules

The argument is:

• If a market is competitive, no individual trader can influence prices.
• If no individual trader can influence prices, the market has many traders.
• A survey shows that the number of traders in the market is large.
• Conclusion: Therefore, the market is competitive.

To assess the validity:

• Break down the statements into logical propositions.
• Use inference rules (like Modus Ponens, Modus Tollens) to determine whether the conclusion follows logically.

Would you like a detailed truth table solution or an explanation of any specific part? Let me know if you need help with the logic steps!

Related Questions

1. How do you construct a truth table for complex logical statements?
2. What are the common rules of inference, such as Modus Ponens and Modus Tollens?
3. What is the difference between tautology, contradiction, and contingency in propositional logic?
4. How do inverse, converse, and contrapositive relate to conditional statements?
5. What are some real-world applications of propositional logic?

Tip: When constructing a truth table for complex statements, start by evaluating smaller subexpressions step-by-step to avoid confusion.