Which of the following statements are true or false? Give reasons for

Question

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

a) The set {S ∈R :x² - 3x + 2 = 0}is an infinite set.

b) The greatest interger function is continuous on R.

c) $\frac{d}{dx}\left[\int_{3}^{e^x}\ln t\,dt\right]=xe^x-\ln 3.$

d) Every integrable function is monotonic.

e) a⊕b = a + b defines a binary operation on Q, the set of rational numbers.

15 Feb 2025

Answer :

Word Count : 431

Let's go through each statement one by one: ### a) The set $ \{ S \in \mathbb{R} : x^2 - 3x + 2 = 0 \} $ is an infinite set. - False. The equation $ x^2 - 3x + 2 = 0 $ is a quadratic equation that can be factored as $ (x - 1)(x - 2) = 0 $, giving two solutions: $ x = 1 $ and $ x = 2 $. Therefore, the set contains only two elements, which is a finite set. Hence, the statement is false. ### b) The greatest integer function is continuous on $ \mathbb{R} $. - False. ________ _________ _________ ___ ____ ____ ______ ________ ___ _______.
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