Key Concepts and Formulas

• Negation of a Disjunction: The negation of a statement of the form "p or q" (denoted $p \vee q$) is "not p and not q" (denoted $\sim p \wedge \sim q$). This is De Morgan's Law: $\sim(p \vee q) \equiv \sim p \wedge \sim q$.
• Negation of a Statement: The negation of a statement reverses its truth value. For example, the negation of "x is an integer" is "x is not an integer".

Step-by-Step Solution

Step 1: Identify the component statements. Let $p$ be the statement "$\sqrt{5}$ is an integer" and let $q$ be the statement "5 is irrational." The given statement is $p \vee q$.

Step 2: Express the negation of the given statement using symbolic logic. We want to find the negation of $p \vee q$, which is $\sim(p \vee q)$.

Step 3: Apply De Morgan's Law. Using De Morgan's Law, $\sim(p \vee q) \equiv \sim p \wedge \sim q$.

Step 4: Translate the symbolic negation back into words. $\sim p$ is the statement "$\sqrt{5}$ is not an integer." $\sim q$ is the statement "5 is not irrational." Therefore, $\sim p \wedge \sim q$ is the statement "$\sqrt{5}$ is not an integer and 5 is not irrational."

Common Mistakes & Tips

• Remember De Morgan's Laws. It's a common mistake to incorrectly negate a disjunction.
• Pay close attention to the wording of the statements. "Not irrational" is equivalent to "rational". However, sticking directly to "not irrational" avoids potential confusion.

Summary

We are asked to find the negation of the statement "$\sqrt{5}$ is an integer or 5 is irrational." First, we represent the two component statements as $p$ and $q$ respectively. The given statement is $p \vee q$. The negation of this statement is $\sim(p \vee q)$, which, by De Morgan's Law, is equivalent to $\sim p \wedge \sim q$. Translating back into words, this is "$\sqrt{5}$ is not an integer and 5 is not irrational."

Final Answer The final answer is \boxed{$\sqrt 5$ is not an integer and 5 is not irrational.}, which corresponds to option (A).