Key Concepts and Formulas

• Logical Equivalence: Two statements are logically equivalent if they have the same truth value for all possible truth values of their components.
• Biconditional ($\leftrightarrow$): $P \leftrightarrow Q$ is true if and only if both P and Q are true or both P and Q are false.
• Negation ($\sim$): $\sim P$ is true if P is false, and false if P is true.
• Disjunction ($\vee$): $P \vee Q$ is true if either P or Q or both are true.

Step-by-Step Solution

Step 1: Express the given statements in symbolic form.

We are given:

• p: x is an irrational number
• q: y is a transcendental number
• r: x is a rational number iff y is a transcendental number

We can rewrite r as: "x is rational if and only if y is transcendental." Since "x is rational" is the negation of "x is irrational", we can write "x is rational" as $\sim p$. Therefore, $r \equiv (\sim p \leftrightarrow q)$.

Step 2: Analyze Statement 1: r is equivalent to either q or p.

Statement 1 claims: $r \equiv q \vee p$. Since we know $r \equiv (\sim p \leftrightarrow q)$, we need to check if $(\sim p \leftrightarrow q) \equiv q \vee p$.

Step 3: Analyze Statement 2: r is equivalent to $\sim \left( {p \leftrightarrow \sim q} \right)$.

Statement 2 claims: $r \equiv \sim (p \leftrightarrow \sim q)$. Since we know $r \equiv (\sim p \leftrightarrow q)$, we need to check if $(\sim p \leftrightarrow q) \equiv \sim (p \leftrightarrow \sim q)$.

Step 4: Construct the truth table for p, q, r, and the expressions in Statement 1 and Statement 2.

We will construct a truth table with columns for p, q, $\sim$p, $\sim$q, r($\sim p \leftrightarrow q$), $q \vee p$, $p \leftrightarrow \sim q$, and $\sim (p \leftrightarrow \sim q)$.

Step 5: Evaluate Statement 1 using the truth table.

Comparing the columns for r ($\sim p \leftrightarrow q$) and $q \vee p$, we see that they are not the same. Specifically, when p is True and q is True, r is False and $q \vee p$ is True. When p is False and q is False, r is False and $q \vee p$ is False. Therefore, Statement 1 is false.

Step 6: Evaluate Statement 2 using the truth table.

Comparing the columns for r ($\sim p \leftrightarrow q$) and $\sim (p \leftrightarrow \sim q)$, we see that they are not the same. Specifically, when p is True and q is True, r is False and $\sim (p \leftrightarrow \sim q)$ is True. When p is True and q is False, r is True and $\sim (p \leftrightarrow \sim q)$ is False. When p is False and q is True, r is True and $\sim (p \leftrightarrow \sim q)$ is False. When p is False and q is False, r is False and $\sim (p \leftrightarrow \sim q)$ is True. Therefore, Statement 2 is false.

Common Mistakes & Tips

• Carefully construct the truth table, paying attention to the order of operations and the definitions of the logical connectives.
• Double-check your truth table entries to avoid errors.
• Remember that two statements are equivalent if and only if their truth tables are identical.

Summary

We analyzed Statement 1 and Statement 2 by expressing them in symbolic form and constructing truth tables. By comparing the truth tables for r and the expressions in Statement 1 and Statement 2, we found that neither statement is true. Therefore, both Statement 1 and Statement 2 are false.

Final Answer

The final answer is \boxed{Statement − 1 is false, Statement − 2 is false}, which corresponds to option (A).