Let's break down your questions step by step: --- ### (a) **Negation** **Statement:** "I have more than 30 close friends." **Negation:** "I have 30 or fewer close friends." (Equivalently, "I have at most 30 close friends.") **Explanation:** - "More than 30" means the number of close friends > 30. - The negation is the opposite: number of close friends ≤ 30, which can be phrased as "30 or fewer" or "at most 30." - We do *not* say "It is not the case that...", as requested. --- ### (b) **Truth of Statements** #### (i) **Statement:** "\(x^2 - 4 = 0\) has at least one real root and \(f(x) = |x|\) is differentiable." Let's analyze both parts: - **Part 1:** \(x^2 - 4 = 0\) Solve: \(x^2 - 4 = 0 \implies x^2 = 4 \implies x = 2\) or \(x = -2\). Both are real numbers. So, this equation **has at least one real root** (in fact, two). - **Part 2:** \(f(x) = |x|\) is differentiable. Recall: - A function is **differentiable** at a point if its derivative exists there. - \(f(x) = |x|\) is differentiable everywhere **except at \(x=0\)**, because the left and right derivatives at 0 are not equal: - For \(x>0\), \(f'(x)=1\). - For \(x<0\), \(f'(x)=-1\). - At \(x=0\), the derivative does **not** exist. **Therefore, \(f(x) = |x|\) is NOT differentiable (everywhere on \(\mathbb{R}\)), since at \(x=0\) it fails.** - **"And" Statement:** The statement uses "and", so both parts must be true for the whole statement to be true. - Part 1: TRUE - Part 2: FALSE **So, the whole statement is FALSE.** --- #### (ii) **Statement:** "If \(x^2 + 4 = 0\) has at least one real root, then \(f(x) = |x|\) is differentiable." This is an **implication** ("If P, then Q"): - \(P\): \(x^2 + 4 = 0\) has at least one real root. - \(Q\): \(f(x) = |x|\) is differentiable. Let's check \(P\): - \(x^2 + 4 = 0 \implies x^2 = -4\) - There is **no real solution** for \(x^2 = -4\), since the square of a real number can never be negative. So, \(P\) is **FALSE**. Recall the **truth table of implication** (\(P \implies Q\)): - If \(P\) is FALSE, the implication is **TRUE** regardless of the truth value of \(Q\). **Therefore, the statement is TRUE.** --- ### **Summary Table** | Part | Truth Value | Explanation | |---------------|:------------:|---------------------------------------------------------------------------------------------------------------------------------------------| | (i) | FALSE | First statement true, second false, "and" means whole statement is false. | | (ii) | TRUE | "If...then..." with a false hypothesis is always true in logic. | --- #### **Key Concepts Used:** - **Negation:** Opposite of "more than 30" is "30 or fewer." - **Differentiability:** \(f(x) = |x|\) is not differentiable at \(x=0\). - **Implication in Logic:** "If \(P\) then \(Q\)" is true whenever \(P\) is false, regardless of \(Q\). - **Roots of Quadratic Equations:** Real roots exist if the discriminant is non-negative. --- **If you need further explanation on any part, let me know!**