9. Prove that there are as many positive real numbers as there are real numbers.

Q: How many positive integers less than 10000 are there which contain at least one 7 or at least one 9…

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Q: Show by contraposition that for any real number x if - 5 < x< 4 then x2+x – 20 < 0

A: For the solution follow the next step.

Q: show that SZ+5 is irrational number

A: If a number cannot be written in ab form, where a and b are integers is known as an irrational…

Q: show that Q (rational numbers) is negligible set.

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Q: write down the proof of the statement " √2 + √5 is not rational" in latex

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Q: 14. Demonstrate that 21 has no primitive root.

A: An integer n≥2 has a primitive root if and only if the the number n is one of the following; 2,4,…

Q: 3. Prove or disprove (x, y are real numbers a) [x- y] = [x] – by] TRUE FALSE because:

A: Introduction: The integer, that is largest and less than or equal to the number a is known as floor…

Q: Prove that if a,b belong R(Real Numbers) then |a|-|b| is lesser or equals to |a-b|

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Q: Show that every positive integer can be written as the product of a square (possibly 1) and a…

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Q: Prove (kindly provide reason/s) that for any real numbers m and n, (-m ) + ( - n ) = - ( m + n)

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Q: Determine whether the statement is true or false. Justify your answer with a proof by Direct Method…

A: The product of any two integers is odd. - True

Q: Prove that 3 is irrational number.

A: Rational Number: A number is of the form of pq, where p, q∈ℝ and gcd(p, q) = 1 Now we have to show…

Q: Prove that 1+6x-x is always a positive real number if x =3+Ai for some real 33 number 4

A: The given number is  1 + 6x - x2

Q: 6. Show that Vz eC i) z-7 is a real number. ii) (z-2) is a real number.

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Q: rove that the difference of any two odd integers is even.

A: Even numbers are of the form 2k. Odd numbers are of the form 2k+1, where k is an integer.  Let a and…

Q: Prove that there is no triangular number which is one less than a multiple of 11.

A: Introduction: A triangular number is a positive integer that is the average of two integers. For…

Q: Prove that there do not exist two rational numbers whose sum is irrational.

A: prove of this question is given below...

Q: 3) Prove that there do not exist two rational numbers whose sum is irrational.

A: To prove: There do not exist any two rational numbers such that their sum is irrational.

Q: Prove that it is not true that the product of two irrational numbers is irrational.

A: To check:   Whether the product of two irrational numbers is irrational or not.

Q: 1. Show that the distance between two numbers in the real number line, given by d(x,y)=|x-yl,…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: What two nonnegative real numbers with a sum of 23 have thelargest possible product?

A: we have to find the two non negative real number with a sum of 23 which will have largest possible…

Q: What two nonnegative real numbers with a sum of 34 have the largest possible product?

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Q: Show that √17 is Irrational number

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Q: 32. Prove that 3 divides n + 2n whenever n is a positive integer.

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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: (2) Prove the statement "Every positive even integer less than 16 can be written as a sum of three…

A: Perfect square: n is a perfect square, if and only if, there exist an integer k such that n=k2.…

Q: let s be positive integer, then (2) + (;) = %3D

A: sC5 = s!/5!*(s-5!) And sC6 = s!/6!*(s-6)!

Q: Prove: “The sum of two even integers is even.”

A: To prove: The sum of two even integers is even. n is an even number if it can be expressed as n =…

Q: Show that 3 + V5 is irrational number

A:

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: 27. Show that if x and y are positive real numbers that have the same first n digits in their…

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Q: : a number is even  if

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Q: 3. Show that (5+ 13)e + (5- i3)-" is a real number if is also a real number.

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Q: 6. Prove that |x.y| = |x|.|y|, where x and y are real numbers.

A:

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

A: Since we know for any real number a,b  If a &lt; b then -b &gt; -a

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: 5!, called 5_______ , is the product of all positive integers from_________ down through________ .…

A: For any positive integer n, n! is the product of all the positive integers  from 1 to n. That is,…

Q: Suppose x is a real number and x is not equal to zero. Prove that if ((x^1/3)+5)/((x^2)+6)=1/x, then…

A: Given that, x is a nonzero real number, and 8 is not a root of given equation.

Q: 1. Prove that 3 is irrational number.

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Q: Prove the Inequality which states using  : If x ≥ −2 then (2+ x)n ≥ 1 + nx for all natural numbers n

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Q: the sum of any positive integers that is less than 30

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Q: True or false, for any integer x, x^2, - x will always result in an even number

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Q: Give a proof by contradiction that there does not exist an integer which is both even and odd.…

A: Assume an integer x which is both odd and even.

Q: The Archimedean Property says that the set of real numbers have no upper bound. True or False

A: Given a statement , we have to check whether it is true or not. As we know the property of…

Q: 2. Show that for any real number x, if x + 2 > 10, then x? – 14 > 5x. Use direct proof.

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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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Q: Prove that there is not rational number whose square is prime

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Q: 9. Prove that the greatest common divisor of two positive integers divides their least common…

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