**4. Write negations of the following quantified statements:**(a) $\forall x \in \mathbb{R}, \text{ if } |x| > 5 \text{ then } x < -5 \lor x > 5$(b) $\exists a, b \in \mathbb{Z}, \, a | b \land b | a$**Explanation:**- **Statement (a)** involves universal quantification over real numbers ($\mathbb{R}$). It states that for all real numbers $x$, if the absolute value of $x$ is greater than 5, then $x$ is either less than -5 or greater than 5. - **Statement (b)** involves existential quantification over integers ($\mathbb{Z}$). It states that there exist integers $a$ and $b$ such that $a$ divides $b$ and $b$ divides $a$.The task is to find the negations of these statements, which involve logical and set theoretic transformations.

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Description: **4. Write negations of the following quantified statements:**(a) \(\forall x \in \mathbb{R}, \text{ if } |x| > 5 \text{ then } x 5$(b) $\exists a, b \in \mathbb{Z}, \, a | b \land b | a$**Explanation:**- **Statement (a)** involves

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4. Write negations of following quantified statements: (a) VxeR, if |x>5 then x<-5 vx>5 (b) 3a,beZ, a|b ^ b|a

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Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

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**4. Write negations of the following quantified statements:**

(a) $\forall x \in \mathbb{R}, \text{ if } |x| > 5 \text{ then } x < -5 \lor x > 5$

(b) $\exists a, b \in \mathbb{Z}, \, a | b \land b | a$

**Explanation:**

- **Statement (a)** involves universal quantification over real numbers ($\mathbb{R}$). It states that for all real numbers $x$, if the absolute value of $x$ is greater than 5, then $x$ is either less than -5 or greater than 5.

- **Statement (b)** involves existential quantification over integers ($\mathbb{Z}$). It states that there exist integers $a$ and $b$ such that $a$ divides $b$ and $b$ divides $a$.

The task is to find the negations of these statements, which involve logical and set theoretic transformations.

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