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

Q: 74. Prove that for any real numbers a and b, |b − a|=|a − b|.

A: Given; a and b are any real numbers Consider, L.H.S=b-aWe can write it as,=b+-a=-1-b+a=-1a-b…

Q: Find all values of c such that the domain of is the set of all real numbers.

A:

Q: . Let a be a positive number. Prove that for each real number x there is an integer n such that na <…

A:

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 or disprove the statement: For all x E R, |x + 1| + |x – 1| > 2.

A:

Q: Prove the statement if it is true, and give a counterexample if it is not true. a. For every real…

A: Both the subparts are solved with explanation.

Q: Prove that for all real numbers r and s, if r is rational, r≠0, and s is irrational, then rs is…

A:

Q: If x and y are real numbers and x>1, prove that there exists a positive integer n such that xn > y

A:

Q: If n is any real number and f(x) = x", then f'(x) = nx"-!

A: Why does ddxxn=nxn-1                 ddxfx=limh→0fx+h-fxh…

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: Given that |f'(x)| < 1 for all real numbers x, show that If(x1) – f(x2)| < \x1 – x2| for all real…

A:

Q: Let æ be a real number. Show that [3r] = [a] + æ + + 3

A: Here we have to show that 3x=x+x+13+x+23

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

A:

Q: Let x be a real number. Prove that x is an integer if and only if [0.891a] + [0.109æ] =

A: Solution: Definitions floor function: It takes real number x as an input and gives as output the…

Q: If R is the set of all real numbers, what do the cartesian products R × Rand R × R × R represent?

A: Cartesian product R × R represents all the coordinates in R2  For example (1,2) , (5,6) coordinates…

Q: For all real numbers x and y, [x]+ [y] = [x + y] – 1. For all real numbers x, [x –1] = [x] – 1. | -

A:

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: Find all values of x such that x² 2+ x + 2! = 0. 3! 4! If no such values of x exist, prove this. + +

A:

Q: 10. Let x be a real number. Prove that at least one of the numbers /2 – x or /2 +x is irrational.

A:

Q: Prove : there exists a real number x such that x^2=3.

A: Let us assume a function f(x)=x2 For x=1, f(1)=12=1  For x=2, f(2)=22=4

Q: Prove that for all real numbers Ex)y= -(1u)

A: We have to prove that for all real numbers (- x)y= - (xy)

Q: Show that if f (x) = anxn + an−1xn−1 +⋯+ a1x + a0, where a0, a1,…, an−1, and an are real numbers and…

A: We are given that f (x) = anxn + an−1xn−1 +⋯+ a1x + a0, where a0, a1,…, an−1, and an are real…

Q: Find functions f and g from the set of positive integers to the set of real numbers such that f (n)…

A: Here, we need to find functions f and g from the set of positive integers to the set of real numbers…

Q: Let {x1, x2,..., xn} C R be a subset of real numbers. Prove that (x1 + x2 + . + an)² < n(x² + x +·.…

A:

Q: if z1-z2 and z1/z2 are both real then prove that two must either real or equal to each other

A: Suppose z1-z2 and z1z2 are both real. We need to prove either z1 and z2 are real or equal to each…

Q: Let a and b be any real numbers such that a 0. Prove that a < b.

A:

Q: Let x and y be positive real numbers. Prove that a + y > 2/xy.

A: See solution below

Q: x-4 f(x) = In %3D x² – 5x + 6)

A: Given:  f(x)= ln x-4x2-5x+6⇒ f(x) =…

Q: Prove that the real part of is constant for -n < 0 < n z+2-i

A: Let  z = a+ib

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

A:

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

A:

Q: If a<b<c<d, then prove that for any real 2,

A:

Q: Theorem: for every real number x, x s |x]. Proof: If x 2 0, by definition |x| = x. Thus |x| > x. If…

A: Contradiction proof : Assuming opposite of our conclusion we show that our given conditions are…

Q: Prove: There exists a positive real number z such that z2=2.

A: Any complex number can be considered as z=x+iy where x,y∈ℝ . Note some result : If z is real number…

Q: Prove that e" > x+ 1, for all positive real numbers a.

A:

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 real number then L2xd =LxJe bx+!]

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

Q: Let R be the set of real numbers and f:R- (1,00[ and g: R→R be defined as f(x) = x² +1 and g(x) = x…

A: Given:- f:R→1,∞      by f(x)=x2+1g:R→R             by g(x)=x+1 Now, f∘gx=fgx f∘gx=fx+1 f∘gx=x2+1+1…

Q: Let a, b be positive real numbers. Prove that a +b< v2(a² +b²).

A: solution:-let a, b are positive real numbersroot mean square(RMS) of a and b =a2+b22arithmetic…

Q: Prove that the number V7 lies between 1 and 2.

A:

Q: Prove or Disprove. There exists a unique real number r such that r ≤ p for all positive real…

A: This question related to real analysis's topic real number system.

Q: f(x) = x+3x−2x+3x−2 and g(x) = x−1x+1x−1x+1 . The domain of f ∘∘ g is the set of all real numbers…

A: Use fog(x) =f(g(x))

Q: . The operation * is defined over the set R of real numbers by a *b = a+b+2ab. What is the identity…

A: We will find out the required value.

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

A:

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

A:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.