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 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.

04 Sep 2025

Answer :

Word Count : 431

(a) False. The set is meant to be $\{x\in\mathbb{R}: x^{2}-3x+2=0\}$. Factoring gives $(x-1)(x-2)=0$, so the real solutions are $x=1$ and $x=2$ only. Hence the solution set contains exactly two elements and is therefore finite, not infinite. In calculus terms, the quadratic polynomial intersects the $x$-axis at two points; there is no continuum of roots, so the zero set cannot be infinite. (b) False. The greatest integer (floor) function $\lfloor x\rfloor$ is piecewise constant with jumps at every integer. For any noninteger $x$, there exists an open interval on which $\lfloor x\rfloor$ is constant, so the function is continuous on each $(n,n+1)$. However, at an integer $n$, the ______ ___ _________ _________ ______.
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