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 an integer. Specify what type of a proof you use.

Q: Prove by contradiction that for any positive two real numbers, x and y, if x·y ≥ 100 then either x…

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Q: Prove that if x is a positive real number, then LV L]] = [V]

A: We have to prove the given equation for all positive real numbers x .

Q: Prove that for every real number b, where b is not equal to 0,= b.

A: Given

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: How to prove that a formula A is valid iff ¬A is unsatisfiable?

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Q: (e) For any nonzero real number z, prove Σ''-1 1-z"

A: The objective is to prove that for any nonzero real number x, ∑i=1nx-1xi=1-x-n

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: Prove that if ƒ = {-2 if x is irrational , then f € R on [a, b].

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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: Suppose x and y are any integers. Prove that if x and y are odd, then x+ 5y² is even

A: Suppose x and y are integers. Prove that if x and y are odd, then x+5y2 is even.

Q: In general, if f(x) is real and continuous in the interval from x to xy and f(x1) and f(xu). between…

A: Since you have asked multiple question, as per company guidelines we will solve the first question…

Q: For any positive integer n, prove that if f(x) = xn, then f′(x) = nxn-1.

A: Given: For any positive integer n,prove that if f(x) = xn, then f′(x) = nxn-1.

Q: Prove that for all positive real a we have 22 +5 20x4 2 4/5. 24 x2 +5

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Q: Find an expression for sup in terms of inf E, sup E, inf F, sup F and prove it. 2F

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Q: Prove that for every real number x, if x > 1 then there is a real number y such that y+ý = 2x.

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Q: Prove f:N - N. F(X) = 5x is injoctive.

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Q: For non-negative integer n, let the function fn be given by (Picture)   (where n! = 1 · 2 · · · n,…

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Q: Prove that x2 + 3x + 3 &gt; 0 for all x.

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Q: Prove that if æ is a real number, then [2a] = [x] +

A: If x is a real number then  to prove- 2x=x+x+12 Greatest integer function is denoted by [x] for any…

Q: prove for any 2 non negative integers x and y, where x>y, if their gcd IS Y, THEN THEIR LCM is x

A: The objective is to show that LCM(x,y)=x, given GCD(x,y)=y; where x and y are non-negative integers…

Q: Show that if f1(x) isΘ(g1(x)), f2(x) isΘ(g2(x)), and f2(x) ≠ 0 and g2(x) ≠ 0 for all real numbers x…

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Q: Given x > 0 and a real number y, there is an n in N such that nx > y.

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Q: Let a and b be positive integers such that ax + by = 7 for some integers x and y. Then %3D

A: We have to choose  current answer of given statement.

Q: Prove that x2 + 3x + 3 ≥ 0 for all x.

A: We have been given the quadratic equation  x2+3x+3

Q: Verify that the function satisfies the three hypotheses of Rolle as a comma-separated fist.) x) = 3x…

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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: Verify that the hypotheses of Rolle's Theorem are satisfied for f (x) = x – 6x + 5 on the interval…

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Q: Prove that, for all real numbers r and all integers k and m, (O()-OC) r- k %3D

A: To prove the below statement for real  number r and all the integers k and m. rmmk=rkr−km−k

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: Let x be a real number. Prove that there is a positive integer n such that the distance between nx…

A: Let x be a real number  To prove there is a positive integer n such that the distanc between nx and…

Q: Prove: |a| &gt;= 0  for all a's in the reals.

A: It is known that the absolute value is the distance of the point or number from the origin.

Q: Prove that if x is a real number, then [2x] = [x] +

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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: Prove that =0

A: ∑i=0ne-λλxx!=1n!∫λ∞ e-xxn    ...............(1) We will proof this by using induction on n. Here,…

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

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Q: Show that if fi(x) is O(g1(x)), f2(x) is O(92(x)), and f2(x) # 0 and g2(x) # 0 for all real numbers…

A: The big theta notation is defined as follows: We can say that fx=Θgx if there exist positive real…

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: Let x be an irrational number. Prove that there exists an irrational number y such that xy is a…

A: Let x be an irrational number. The main objective is to prove that there exists an irrational number…

Q: Prove that if x is a real number, then [2r] = [r] + r+

A: Let us consider x=n+∈     (Because x is a real number) where n is integer part and ∈ is fractional…

Q: Prove that if x is real number then L2xd =LxJe bx+!]

A: We will prove the expression by two  steps taking real number x in two form.

Q: Show that if f1(x) is O(g1(x)), f2(x) is O(92(x)), and f2(x) # 0 and g2(x) # 0 for all real numbers…

A: Big theta notation is nothing but an order notation that is used to compare the the growth rates or…

Q: Let x, y and z be real numbers. If x > y and y = z, then x > z. For any real numbers x and y, with x…

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

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Q: . Prove that |x|+ |x + ;] = |2x| for any real number .

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Q: Suppose that o is differentiable. Prove the identity

A: follow next step

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: Find all values of x such that 2+x + 2! ... = 0. + 4! If no such values of x exist, prove this.

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Q: m above, prove that for every real number c>0, there is a 1/N < c.

A: In every real number c, there is integer N which is N &gt; c.

Q: Let x +1 be a real number. Prove that for all n E N, x" – 1 - + x" +...+ x2 + x + 1. x – 1

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Q: 3. Show that for any real number x, if x2 is odd, then x is odd.

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