Which of the following statements are true or false? Give reason

Question

Which of the following statements are true or false? Give reasons for your answers in the form of a short proof or counter-example, whichever is appropriate:

i) Every infinite set is an open set.

ii) The negation of p^~qis p→q.

iii) -1 is a limit point of the interval ]-2,1].

iv) The necessary condition for a function f to be integrable is that it is continuous.

The function f: R→R defined by f(x) = |x-2|+|3-x is differentiable at x = 5.

04 Sep 2025

Answer :

Word Count : 278

i) Every infinite set is an open set. False. Consider the set of integers $\mathbb{Z} \subset \mathbb{R}$. It is infinite but not open in $\mathbb{R}$, because for any integer $n$, any neighborhood $(n-\epsilon, n+\epsilon)$ contains non-integer points, which are not in $\mathbb{Z}$. --- ii) The negation ________ __________ __________ _________ ____ _________ __________.
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