1. Prove that R\ Q is dense in R i.e. prove that any interval ]a, b[, a, b € R, a < b, contains at least one irrational number. Hint: Use the following two facts: ● Qis dense in R, √2 is irrational.

Q: 2. Show that the limit set of the set of rational numbers, i.e. Q', is the set of real numbers R,…

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Q: Let A be a non-empty set of real numbers which has a upremum and let B= {x : - x e A}. Show that B…

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Q: Prove that 1 1 16 ≤ (a+b+c+d) = + = + b + ax (²=²+ + + + + + + ² + 1 = 0) d for all positive numbers…

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Q: 5.)Write the aua mented matrix for the syshem X, +2X,-X, tXc =l =マ+ t bulo

A: We have to solve given problem:

Q: +a2+C √ √√a² + u²du = ½ (u√u² + a² + a²]n]u + √u² + a² 1) + T 114

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Q: Which of the following propositions are true? Choose all that apply.  A. ∀n ∈ Z,2n is even B. ∃n ∈…

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Q: Prove there are no natural numbers a and b such that √a+b = √a + √b.c

A: Assume such a,b exists and prove by contradiction

Q: 5. Let F= [3,2,1] and r= [2,1,–3]. Compute the following: %3D a) Proj „F. b) Orth rF = F – Proj „F.…

A: We know that  ProjrF=r.F∥r∥2r Now, r.F=23+12-31=5 ∥r∥2=r,r        =2,1,-3,2,1,-3        =22+11-3-3…

Q: Exercise 5.11. Assume the universe is the set of real numbers R. Are the following propositions…

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Q: 62. Let D be the domain bounded by y = x2 +1 and y = 2. Prove the incquality 4 20 (x² dA 3

A: The domain is y=x2+1 and y=2. Sketch the domain D as shown below.

Q: 1. Use the proof that m* ([a,b]) =b-a for real numbers azb. Use this to prove the same is true for…

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Q: Prove that the infimum and the supremum of the interval (a,b) are a and b, respectively

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Q: Prove that if 0 < a < 1 and b = 1-√1-a, then 0 < b < a.

A: To prove If  and  then

Q: th 6) Let t, be the k" triangular number. Prove that t, + 4 k + 10 is k k also triangular.

A: It is given that tk is a triangular number that means tk=k(k+1)2. To prove: tk+4k+10 is triangular.…

Q: The ancient Greeks had the following definition: two lengths (positive numbers) a,b are com-…

A: (a) The negation of Pythagoras’ claim can be stated in simple terms as, “there exist at least one…

Q: 1. Let a, b, c E R, and w' = 1, with w # 1 has strictly positive imaginary part. If a, b, c are not…

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Q: Let S be the line segment connecting i to 1, oriented either way. Show that 1 1/2 d S dz ≤ 2√2

A: Sol.    Given that   S is the line segment connecting i to 1.i.e.,    S={(x,y): x+y=1,  0x,y1} .…

Q: - Give the quadratic function fle)= x?-5x Compute flath) -f(a) | flath)-f(a)

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