Key Concepts and Formulas

• Logical Equivalence of Biconditional: $p \leftrightarrow q \equiv (p \rightarrow q) \wedge (q \rightarrow p)$
• Logical Equivalence of Conditional: $p \rightarrow q \equiv \sim p \vee q$
• De Morgan's Laws: $\sim(p \wedge q) \equiv \sim p \vee \sim q$ and $\sim(p \vee q) \equiv \sim p \wedge \sim q$
• Negation of conjunction and disjunction: $\sim (p \wedge q) \equiv (\sim p \vee \sim q)$ and $\sim (p \vee q) \equiv (\sim p \wedge \sim q)$

Step-by-Step Solution

Step 1: Express the biconditional using conditionals. We start with the given expression $x \leftrightarrow \sim y$ and use the equivalence $p \leftrightarrow q \equiv (p \rightarrow q) \wedge (q \rightarrow p)$. $x \leftrightarrow \sim y \equiv (x \rightarrow \sim y) \wedge (\sim y \rightarrow x)$ This step breaks down the biconditional into a conjunction of two conditional statements, which will be easier to manipulate further.

Step 2: Express the conditionals using disjunctions. We use the equivalence $p \rightarrow q \equiv \sim p \vee q$ to rewrite the conditional statements as disjunctions. $(x \rightarrow \sim y) \wedge (\sim y \rightarrow x) \equiv (\sim x \vee \sim y) \wedge (\sim (\sim y) \vee x)$ $(\sim x \vee \sim y) \wedge (\sim (\sim y) \vee x) \equiv (\sim x \vee \sim y) \wedge (y \vee x)$ This step converts the implications into disjunctions, preparing us for applying De Morgan's laws after negation.

Step 3: Negate the entire expression. We negate the expression we obtained in the previous step: $\sim (x \leftrightarrow \sim y) \equiv \sim [(\sim x \vee \sim y) \wedge (y \vee x)]$ This is the core step where we apply the negation to the entire expression.

Step 4: Apply De Morgan's Law. We apply De Morgan's Law to the conjunction: $\sim (p \wedge q) \equiv \sim p \vee \sim q$. $\sim [(\sim x \vee \sim y) \wedge (y \vee x)] \equiv \sim (\sim x \vee \sim y) \vee \sim (y \vee x)$ This step distributes the negation across the conjunction, turning it into a disjunction of negations.

Step 5: Apply De Morgan's Law again. We apply De Morgan's Law to each of the disjunctions: $\sim (p \vee q) \equiv \sim p \wedge \sim q$. $\sim (\sim x \vee \sim y) \vee \sim (y \vee x) \equiv (\sim (\sim x) \wedge \sim (\sim y)) \vee (\sim y \wedge \sim x)$ This step distributes the negations across the disjunctions, resulting in conjunctions.

Step 6: Simplify the expression. We simplify the double negations: $\sim (\sim p) \equiv p$. $(\sim (\sim x) \wedge \sim (\sim y)) \vee (\sim y \wedge \sim x) \equiv (x \wedge y) \vee (\sim y \wedge \sim x)$ This step removes the double negations, leading to a simpler and more recognizable form.

Step 7: Rearrange the terms for clarity (Optional). We can rewrite $(\sim y \wedge \sim x)$ as $(\sim x \wedge \sim y)$ because conjunction is commutative. This step is optional but can help match the result to the provided options. $(x \wedge y) \vee (\sim y \wedge \sim x) \equiv (x \wedge y) \vee (\sim x \wedge \sim y)$

Common Mistakes & Tips

• Remember the correct De Morgan's Laws. It's easy to mix up conjunction and disjunction.
• When negating expressions, work step-by-step to avoid errors. Negate the outermost operator first and then work inwards.
• Be careful with the order of operations. Parentheses are crucial for ensuring the correct interpretation of the expression.

Summary

We started with the biconditional expression $x \leftrightarrow \sim y$, expressed it as a conjunction of conditionals, and then negated the entire expression. By applying De Morgan's Laws and simplifying, we arrived at the equivalent expression $(x \wedge y) \vee (\sim x \wedge \sim y)$.

Final Answer

The final answer is $\boxed{\left( {x \wedge y} \right) \vee \left( { \sim x \wedge \sim y} \right)}$, which corresponds to option (A).