Let A denote the set of positive rational numbers whose square is less than 2: A = {q ∈ Q : q > 0 and q2 < 2}  Show that A has no greatest element.

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

A:

Q: Show v2 is not rational and then conclude that the following set in Q does not have a supremum; A =…

A: We will show root 2 is irrational by first supposing that it is a rational and in the end showing…

Q: Find the maximum value of a, x, +a,x, + .+ax to the condition that x +x, +. + x =1 for given…

A: We need to find the maximum value of a1x1+a2x2+a3x3+...+anxn to the condition that…

Q: Exercise 9. We can find a non-constant polynomial that is bounded over the set of complex numbers.…

A: This is false

Q: Which of the positive integers coming from the set{1, 2, 3, 4} are upper bounds for the zeros of the…

A: The given polynomial is Px=x4-4x2-3

Q: Prove that if x is a rational number andy is an irrational number, then x + y is an irrational…

A: As you have asked the problem ( b) only. I have solved this problem in two steps by showing the…

Q: 2. The set of all rational numbers in the interval (0, 1).

A:

Q: 3) Tusuer is. C. ☆ Should bein lo0 Polymomiaul-defined as 23 . If 2 + 3i is a zero of a polynomial…

A: To Determine: If 2+3i is a zero of a polynomial function with real coefficients. Determine the…

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: Show that the Archimedean Property is a consequence of the assertion that every interval (a,b)…

A: Archimedian property : For every positive real number c there exist a natural number n such that…

Q: 10. Show that the polynomial p(x)=x²-3x³-x² - 4x-20 has at least two distinct roots (Hin use the…

A: We have given,   f(x) = x4 - x2 - 4x - 20

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

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

Q: Find the number of nonnegative integer solutions to x₁ + x₂ + x3 + x4 + x5 ≤ 23. Find the number of…

A: We have to find the number of non negative integer solution of Note: According to the guidelines we…

Q: If m and n are rational numbers such that √m +√n = √5+√24 ,calcule a possible value for m² + n²

A: Given that √m +√n = √5+√24 To calcule a possible value for m² + n²

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: If a, b, and c are non-negative real numbers that satisfy a + 3b + 2c = 3 and 3a + 3b + c = 4.…

A: We will first solve a + 3b + 2c = 3 and 3a + 3b + c = 4. So,    a+3b+2c=33a+3b+c=4Substitute…

Q: Indicate the least integer n such that (3x5 + x²)4 = 0(x¹). E

A:

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: What is the largest number x such that x<200 and x has exactly 4 factors?

A: Factors:    Number that are multiplied together to get a product                              Factor…

Q: Prove that for every real number x, 3x - 6 ≤ x iff |x-1| ≤

A: The objective is to prove that , where  is a real number.The inequality  is equivalent to .

Q: 15) -3 + V2, -1+ Vi A) -3 - V2, -1 - VT B) 3+ V2, 1+ vT C) None DI 3

A:

Q: Whether the polynomial x² + 2x + 2 over Z3, Z5, and Z7 is irreducible or not and state all zeros.

A:

Q: 7. Prove that is rational if and only if 3x + 7 is rational.

A:

Q: (i) Show that √6 + √7 is an algebraic number.

A: We are answering only the first question as per the guidelines. Please repost the other question to…

Q: show that there doesnt exist a rational numbert such that t2 = 3

A: According to the given information, it is required to show that there does not exist any rational…

Q: a. The remainder of P(x) = 2x' +mx? –7 when it is divided by x+2 is 6 greater than when it is…

A:

Q: Prove by reductio ad absurdum that there does not exist a minimum rational number greater than zero.

A: The proof by method of reductio and absurdum mean proof by contradiction . That is , you have a…

Q: Suppose X₁ ~ N(6, 1) and X₂ ~ N(7,2), and X₁ is independent of X₂. Find P(X₁ < X₂) and P(X₁ < X₂ +…

A: We know linear combination of normal distribution is also a normal distribution. Therefore any…

Q: Among the integers 1 ≤ a ≤ 17, which are quadratic residues of 17 and quadratic nonresidues of 17?

A: We have to find among the integers 1 ≤ a ≤ 17, which are quadratic residues of 17 and quadratic…