provide detailed proofs for the following a) Let y ∈ R. Prove: If y ∈ {2x + 3 : x ∈ (−∞, 1)}, then y − 1 < 4. b) Let r ∈ R. Prove: r3 − r2 < 0 if and only if r 6= 0 and r < 1. Prove: There exists n ∈ Z such that n2 + n = 12. c) Define f : R → R by f(x) = x2 − x. Prove: f(x) < 0 for some x ∈ R.

provide detailed proofs for the following a) Let y ∈ R. Prove: If y ∈ {2x + 3 : x ∈ (−∞, 1)}, then y − 1 < 4. b) Let r ∈ R. Prove: r3 − r2 < 0 if and only if r 6= 0 and r < 1. Prove: There exists n ∈ Z such that n2 + n = 12. c) Define f : R → R by f(x) = x2 − x. Prove: f(x) < 0 for some x ∈ R.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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