State whether the following statements are true or false. Ju

Question

State whether the following statements are true or false. Justify your answers with a short proof or a counterexample.
i) The negation of $p \lor (\sim q)$ is $q \to p$ .
ii) $\mathbb{Q} \setminus \mathbb{N}$ is countable.
iii) The equation $x^3 - 2x + 3 = 0$ has a real root between -2 and 1.
iv) Every increasing sequence has a convergent subsequence.
v) The function defined as
$$f(x) = \begin{cases} \frac{\sin 2x}{x}, & \text{if } x \neq 0 \\ 3, & \text{if } x = 0 \end{cases}$
has a removable discontinuity.
vi) An integrable function can have finitely many points of discontinuity.
vii) The series $\sum_{n=0}^{\infty} (-1)^n \frac{1}{2^n}$ is divergent.
viii) The function f defined by $f(x) = \sin x(1 + \cos x), \ x \in [0, 2\pi]$ , has local maximum at $x = \frac{\pi}{3}$ .
ix) Every constant function on [a, b] is Riemann Integrable.
x) The function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = |x - 2| + |x - 1|$ is differentiable at $x = 3$ .

20 Jan 2026

Answer :

Word Count : 188

i) False. The negation of an equation (f(x) = g(x)) is (f(x) \neq g(x)), not the same equation. ii) True. The set of rational numbers, often represented by "equation" in countability contexts, is countable. iii) True. By _______ ___ _____ _______ ______ _____ _____ _______ ___ _______ ________ _______.
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