Question-01 (a) Prove that if x is a real number, then [2x] = [x] + [x +. Question-01 (b) Prove or disprove that [x+ y] = [x]+ [y] for all real numbers x and y.

Q: -. For any integer m and any real number x, if x is not an integer, then [x]+[m - x] = m-1.

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Q: Prove this theorem Let a,b,c be an element of real numbers where a is not equal to zero. The…

A: Use the provided information.

Q: ∀ (x,y) ∈ ℝ^2 (x and y are real numbers), then | |x| - |y| | ≤ |x - y|

A: Case 1: Suppose x≥0 and y≥0. By definition of modules function, we have x-y=x-y≤x-y. Case 2: Suppose…

Q: Using proof by contradiction show that for any positive real numbers x, y that x +y > 2/xy.

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Q: Prove that the product of a negative real number and a positive real number is negative.

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Q: (n-1)! Use Euler's theorem to prove that if n is an odd integer, then n divides 2 -1.

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Q: Proof that 3√2 is an irrational.

A: Given query is to prove that 3√2 is an irrational.

Q: Third degree, with zeros of -3,-1, and 2, and passes through the point

A: We have to find polynomial

Q: EVALUATION: Prove the following. 1. Using direct proof, show that "if x and y are sum of two integer…

A: Note: Since you have posted multiple questions we will solve the first question for you. If you want…

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

A: Yes, the product of  even integer and any integer equal to even

Q: Find two positive integers such that the sum of the first number and four times the second number is…

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Q: Using contradiction prove or disprove: For all real numbers x, (√2 + x) is irrational or (√2 − x) is…

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Q: Direct proof that not every root of polynomial is a rational numbe

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Q: Using indirect proof:  If x is a real number and x³-x²+x-1=0, then x=1.

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Q: Let r be any real number, where z is NOT an integer. Find a general rule (for any non-integer real…

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Q: see the equation as attached here

A: Solution: Consider f be an even function then by definition its nth Maclaurin polynomial contains f…

Q: uestion 11 3! x (x - 1 = 0),where domain consists of all integer numbers True False

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Q: Find a one-to-one correspondence between [0, 1] and (0, 1). Find a one-to-one correspondence between…

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Q: do i prove this

A: Introduction: A triangle could be a 3-edged polygonal shape with three vertices. it's an elementary…

Q: 3. Irrational Prove using a proof by contradiction that the product of a nonzero rational number and…

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Q: Using only the definition of rational and irrational numbers, show that if (n + 3)² is irrational,…

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Q: 75. DISCUSS: Polynomials of Odd Degree The Conjugate Zeros Theorem says that the complex zeros of a…

A: To explain: The Fact that the polynomial with real coefficient and odd degrees has at least one real…

Q: 53-56 Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

A: We can prove that f(x) has a root between (a, b) if we can shown that f(a) and f(b) are different in…

Q: (a) Show that a polynomial of degree 3 has at most threereal roots.(b) Show that a polynomial of…

A: a) Let a polynomial with leading coefficient 1 and with degree 3 has more than 3 real roots. For…

Q: Using indirect proof:  Prove: If x2-4=0, then x = 2 or x = -2.

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Q: 28. Proving a Factorization Show that x + y is a factor of -1+ yan-1 for all natural numbers n.

A: Let the given statement be P(n)             P(n) = x2n-1 + y2n-1We check P(n) for n=1 P(1) =…

Q: Buniakovski26 inequality) If neN and a,a2, ... ,an, b,b2,..,bn are real numbers, prove that k%3D1…

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Q: Prove showing all arguments Pythagoras theorem

A: SOLUTION-

Q: Exercies: 1) Express the polynomial x + 2x? - 5x + 2 in a series of Laguerre polynomials.

A: Given that, x3+2x2-5x+2 is polynomial  We Know that formula for laguerre polynomial is,…

Q: USING DIRECT PROOF   If x is a real number and x³-x²+x-1=0, then x=1

A: We can solve this by using definition of real number

Q: pdf. Every open set in R contains an irrational number. ths O

A: False

Q: True or False Rational numbers and irrational numbers are in the set of real numbers.

A: Known fact: Set of real numbers contains rational numbers and irrational numbers.

Q: if p is an even integer, then p-5 is odd. prove using direct method. show your solution and proving.

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Q: Decimal representing of a rational number cannot be non-terminating and non- repeating.

A: Let x be a terminating and non-repeating decimal number. Find the rational number representation for…

Q: Pre Calculus (not Calculus) Give an example or state none exist. An irrational number between -4…

A: Que:Give an example or state none exist.   1.An irrational number between -4 and -3. 2. A negative…

Q: an+1 - a 1.) Let a +1 be a real number. Prove that a + a² + a³ + • · ·+ a" for all integers n> 1. a…

A: We will use the principle of mathematical induction to prove the given statement. First we will show…

Q: USING DIRECT PROOF:  Prove: If x²-4=0, then x = 2 or x = -2.

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Q: Calculate the following limits using the factorization formula

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Q: Formulate and prove a theorem about which numbers can be represented as the sum of 1000 consecutive…

A: Solution: We know that for any positive integer n, the sum 1+2+3+........+n is given by,…

Q: Using indirect proof: Prove If x²-4=0, then x = 2 or x = -2

A: Indirect proofs are two types : the prove by contrapositive and the prove by contradiction. We shall…

Q: Proof of the quartic formula

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Q: An algebraic number is any real root of an equation ao + a₁x + +anx"=0, where the coefficients a;…

A: We use properties of Countable union of countable sets and set of polynomials.

Q: USING DIRECT PROOF Prove that the product of and even number and an odd number is an even number

A: Prove that the product of and even number and an odd number is an  even number

Q: QUANTIFICATION  True or False  FOR SOME x, x is any REAL number: x^2 + 1 = 0

A: To determine whether x is a real number or not for the below expression. x2+1=0

Q: 5. (10 points) Prove that for all positive real numbers r and y, one has the inequality > 2 (u+ 112…

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Q: Proposition: There exist rational numbers a and b such that ab is rational. Proof: Let a = b = 2.…

A: Non constructive existence proof A proof which indirectly shows a mathematical object exists without…

Q: proof that: x^8 + x^6 - 4x^4 + x^2 + 4 > 0

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Q: Prove using proof of contradiction. Include explanation.   If n is an even integer, then 7n + 4 is…

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Q: Using indirect proof:  Prove that the product of and even number and an odd number is an  even…

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