Q1. 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. (80 words)
- (b)) The greatest integer function is continuous on R. (80 words)
- (c)) d/dx [integral from 3 to x of (e^t / ln t) dt] = xex - In 3. (80 words)
- (d)) Every integrable function is monotonic. (80 words)
- (e)) a*b = sqrt(a+b) defines a binary operation on Q, the set of rational numbers. (80 words)
Key Points:
- A set containing specific solutions to an equation is finite if the number of solutions is finite.
- The greatest integer function has jump discontinuities at all integer points, hence not continuous on R.
- The First Fundamental Theorem of Calculus differentiates an integral ∫(a to x) f(t) dt to f(x).
- Integrability does not imply monotonicity; many non-monotonic functions are integrable (e.g., sin(x)).
Answer: Here are the detailed analyses for each sub-question, determining whether the statements are true or false, along with appropriate justifications.