Suppose that x is a negative real number and that x < 1/x. Prove that x <-1.

Q: Suppose h is continuous on ℝ and h(q) = 0 for every rational number q. Does it follow that h(x) = 0…

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Q: Suppose the function f is defined on a measurable set E and has the property that {x ∈ E | f(x) > c}…

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Q: Show that log, q is irrational for any distinct prime numbers p and q. (Distinct here p# q.) means

A: To Determine :-  logp q is irrational for any distinct prime numbers p and q.

Q: find a Positive integern for which the equality holds %3D

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Q: Prove by contradiction that for any positive two real numbers, x and y, if x · y ≤ 50, then either x…

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Q: Prove the following using the specified technique: Let x and y be two real numbers such that x+y is…

A: Consider the given information. Given that x and y be two real numbers such that x+y is rational. To…

Q: rove that the negative of any even integer is even

A: Assume 2n to be an even integer. 2n = even integer -2n = negative of even integer  Let a = -2n be an…

Q: a. Prove that ex&gt;= 1 + x if x&gt;= 0. b. Use the result in part (a) to show that ex&gt;= 1 + x +…

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Q: Find the domain of f(x)=x² for 0<x<5. O (1,25] O (0,25) O [0,25] O [1,25)

A: Given: The function is fx=x2 for 0≤x≤5. To find: The domain of the given function. The given options…

Q: Suppose there is a real number r with the property that r + 1/r is anodd integer. Prove that r is…

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Q: Suppose h is continuous on ℝ and h(q) = 0 for every rational number q. Does it follow that h(x) = 0…

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Q: Prove that if x is a real number, then ⌊2x⌋ = ⌊x⌋ + ⌊x + 1/2 ⌋.

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Q: Prove that Q(3^(1/3), sqrt(3))= Q(3^(1/6)), where Q is the set of all rational numbers.

A: First observe that,

Q: Show that if ƒ is both even and odd, then ƒ(x) = 0 for every x in the domain of ƒ

A: Given, function f is both even and odd function.

Q: The interval ___________ consists of all real numbers for which x ≤ a.

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Q: Suppose that .3 f(3) = }, and f(x) > 0, and juw <0. < 0. Is it possible that f(2) = . %3D False O…

A: You have asked two different questions. As per the company guidelines we are supposed to solve first…

Q: Let x be a rational number. Prove that if ry is irrational, then y is irrational.

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Q: Find a number that provides a counterexample to show that a given statement is false, For all number…

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Q: If f(x) has odd degree and negative leading coefficient, as x goes to -∞, then f(x) would go to

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Q: show that: (For real nu

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Q: Show that there are an infinite number of rational numbers between any two real numbers x and y with…

A: using Archimedean property to prove above therom...

Q: 60. Prove that for any positive integer n, for every real number x. %3D

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Q: Find the value of x to make a counterexample to the conjecture. Conjecture: For all numbers x,…

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Q: (4) Prove that there is no smallest positive real number.

A: I have to prove that there is no smallest positive real number . Suppose that there is a smallest…

Q: Show that for any real numbers X and y. one of the following two case can happen: Either [x+Y = [×]…

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Q: Provide either a proof or a counterexample for  For all positive real numbers x, x^2 - x ≥ 0

A: According to the given information, it is required to prove the given equality or give a counter…

Q: Prove or disprove that there is a rational number x and an irrational number that xy is irrational.…

A: The given statement is false.

Q: Let x be an element of the real numbers. Prove |x| is greater-than or equal to 0. State any theorems…

A: For any real number x , the absolute or modulus of x is denoted byx and is defined as x=x , if…

Q: Prove or disprove : [-x]=-[x] for any real number x

A: Suppose, [[-x]]=-[[x]] is true for all the real numbers x. Here, [[x]] is the greatest integer…

Q: Show that if x is a real number, then [x] – [x] = 1 if x is not an integer and [x] – [x] = 0 if x is…

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Q: Prove the inequality √1+x≤1+x2 for x&gt;1.

A: Consider the function. Find its derivative.

Q: Prove that x (x-2) is always a negative real number if x 1-Ai for some real number A.

A: The given number is  x(x - 2) x = 1 - Ai

Q: Suppose that 0 &lt;ƒ′(x) &lt; 1/2 for all x-values. Show that ƒ(-1) &lt; ƒ(1) &lt; 2 + ƒ(-1)

A: From the given inequality the behaviour of the function can be determined as.

Q: 1, when x is a rational number. f(x)= 0, when x is not a rational number. Determine if the function…

A: Given:  fx=1, when x is a rational0,  when x is not a rational To determine the function  f is…

Q: Suppose x is a fixed real number with x 2 -1. For n2 1, prove using mathematical induction: (1+ x)"…

A: Here we will show that given statement is true by mathematical induction method.

Q: - Give the domain for k(a) in interval notation. x2 + 3x +2

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Q: 2. Let x be irrational real number. Prove that 2x+3 is irrational

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Q: The function f(x) = 4 for x < 2 and f(x) = 4 for x 2 2 has domain of all the real numbers. To True O…

A: As we know that domain is all the value of input (x-values) for which a function can define. for…

Q: Find a counterexample to show that the following conjecture is false. Conjecture: For all numbers…

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Q: Prove that there is no rational number x such that x2= 3.

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Q: Show that the interval [0, 1] is uncountable.

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Q: Let x be a real number. Prove by contrapositive that if æ + 7x = 90, then x < 4.

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Q: Prove that there are as many positive real numbers as there are real numbers.

A: Here , we have prove that all positive numbers are real numbers.

Q: Prove that if æ is a positive real number, then

A: To prove that if x is a positive real number, then                               x=x

Q: Frngl. prove that x

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Q: For which positive numbers a is it true that a" > 1 + x for all x?

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Q: Prove that if æ is a positive real number, then LV=]] = LVE] %3D

A: Solution: Definition: The Floor function f is a function that takes any real number x and gives…

Q: Prove or Disprove. for any real numbers x and y, |x+y| = |x| + |y|

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Q: Use a direct proof to show that if x is a real number such that (x – 1)? = 0, then x³ –1 = 0.

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