Proficiency #7. [Counterexamples] For each of the two following statements, give a counterexample to show that the state- ment is false. a. (Vr e R)((x > 0) → (x³ > x²)). b. If n E Z, then there exists m e Z such that n = m² mod 3.

Q: 4. Let F = {s+t√2 s,te Q}. Show that F is a subfield of R. You may assume that √2&Q.

A: The objective of the question is to prove that the set F, defined as F = {s + t * sqrt(2) | s, t…

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Q: Find all the values of t so that 9eBt > 0. Ot≥0 00 21/1/20 Ot< 1/1/2 9 -00 < t < 00

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Q: 1 Q6) Let H(x):=k for x=, (k eN) and H(x):=0 elsewhere on [0,1] k Determine whether H(x) eR[0,1 or…

A: Unbounded function is not Riemann integrable.

Q: 3. Prove or disprove the given statement. (a) If x e R, and x + 0, then 2x > x. (b) If n e Z*, then…

A: Given: (a) If x∈R, and x≠0,then 2x&gt;x (b) If n∈Z+, then 2n-1&gt;0

Q: Show that: 8 [J₁(bt) t" eat dt ed 0 2"b" I(n+-) 2 n+- 2 √√π (a² + b²) T

A: To prove the given integral identity, we will use the integral representation of the Bessel function…

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Q: ind all solutions for n EN for which P(n, 4) = 12P(n, 2).

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Q: Proof by contrapositive that for xy EZ, if xy is even then either x or y is even.

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