5. What is the completion of (X, d). where X is the set of all rational numbers and d(xr, y)

Q: B) Let f:R - R be defined by. 3 if x rational number. f(x) = if x irrational number !3! Show that f…

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Q: (e) If x and y are rational numbers, where y +0 and y # -1, then 1+ is also rational. 1

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Q: Suppose h is continuous on ℝ and h(q) = 0 for every rational number q. Does it follow that h(x) = 0…

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Q: Suppose the function f is defined on a measurable set E and has the property that {x ∈ E | f(x) > c}…

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Q: 4. [5] Please find in interval notation the values of x for which this expression is real. Show…

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Q: Prove

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Q: 2. Show that the limit set of the set of rational numbers, i.e. Q', is the set of real numbers R,…

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Q: Let S = { all (x, y) with æ and y rational numbers j. (a) What is the interior of S ? (b) What is…

A: We know interior of the set of rational numbers is empty and boundary of the set of rational numbers…

Q: Suppose x is a positive rational number and x? < 2. Let y = 22. Prove that x < y and y? < 2. X+2

A: We will use basic knowledge of set theory and logic to answer this question.

Q: 4. Use the algebraic description to find the image of AMNO). (x, y) → (x+ 2,y- 3) -3 21

A: From the figure, we see that M(-2,4), N(-3,1) , O(-1,0).

Q: 2.6.10. Take X = R, and find E°, DE, and E for each of the following sets. (a) E %=D {1, 글, 흙, }.…

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Q: 4. Let x, y, and z be real numbers. Prove that if x and y are rational numbers and z is an…

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Q: 1 Find the absolute maxima and minuma x²-2x4+ bunded by y= x², y=t, the closed negion on %3D…

A: Given that, f(x,y)=x2-2xy+y Take the derivative for x, fx=2x-2y Take the derivative for y, fy=-2x+1…

Q: Prove that Q(3^(1/3), sqrt(3))= Q(3^(1/6)), where Q is the set of all rational numbers.

A: First observe that,

Q: D if x,Y,2 inclepndent Variables and z function in X, Y which achieves the eguation find az…

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Q: Prove for all rational number x and y and all irrational number r. If (r+x)/(r+y) is rational then…

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Q: Prove or disprove that there exist rational numbers x, y EQ such that √11: = x+y√5.

A: Prove or disprove that there exist rational numbers x, y  in Q such that √11 = x+y√5.

Q: ind the set af domain in the Rational expres sions. show complete Solution) (5 points)

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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: If D is an open subset of R, which contains all rational numbers lying between 0 and 2, then does V2…

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Q: 1 1. -to prove that x+y|# | x|+ | y| for all real numbers d) Use x = and y = %3D x and y.

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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: Given: "if x is rational and x + 0, then 1/x is rational", can be proven by Direct proof. O True…

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Q: 10. (a) Prove: If x and y are real numbers with x < y, then there are infinitely many rational…

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Q: 1. Let f:R R be defined by {1 if x rational number, 12 if x irrational number f(x) = Show that f is…

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Q: 4cos20 + 2/3 = 0 on the interval[0, 3t]

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Q: Prove that (z - x)(z - y) = (y - x)?. a:= is not a rational number. Hint: Express (x - x) and (z -…

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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: 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: Prove the following statement using a direct proof. If x and y are rational numbers, then 4x + y is…

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Q: 43. Show that there cannot exist rational numbers a, b such that V3 = a+b/2.

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Q: Prove or disprove that there is a rational number x and an irrational number that xy is irrational.…

A: The given statement is false.

Q: Suppose x and y are positive real numbers. If x< y, then x² < y².

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Q: The product of any non-zero rational number and any irrational number is irrational. For all real…

A: To prove the statement by using the method of contradiction, first assume that the conclusion is…

Q: Suppose x is a real number. Prove that if x 2, then there is a real number y such that x = (2y + 1)/…

A: Domain of a Function The domain of a function explains what values the independent variable, X, can…

Q: In Rn , Let R denotes the set of points having only rational coordinates and l its complements ,…

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Q: a. Suppose that r E Q and x E Qº. Prove or disprove: The number (r – x)- E Q°. b. Prove or disprove.…

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Q: If a , b are two rational numbers such that Va + vb is a rational number, then prove that Vab is…

A: We have given that , a  and  b are two rational numbers  such that  a + b  is a rational number. We…

Q: What is the midpoint between z1 = 9i and z2 = 4? (2, 4.5) (-4,-9) (-2,-4.5) (4, 9)

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Q: The base of a three-dimensional figure is bound by the graph x = y° on the interv 1, 2]. Vertical…

A: As it is known that the area of an isosceles triangle is given as 12bh where b is the base and h is…

Q: Let A = 3 21 then A-1 Does not exist O None of these equal to the above Dequal to the above

A: If determinent of metrix is zero then then matrix has no inverse

Q: Let Q denote the set of rational numbers. Show that card(Q)=x0. What is card(R/Q)?

A: Solution:- Step1:-      It is given that ℚ is the set of rational number  then,        we have to…

Q: Show by example: a) that the least upper bound of a set of rational numbers need not be rational. b)…

A: Here we use the density property of real number.

Q: 1+ d(z, y) is complete if and only if (X, p) is complete. Question 5 : What is the completion of…

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Q: prove

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Q: Use intermediate theorem to explain P [x] = x* – x' – 10 has a zero in [2, 3]

A: According to intermediate value theorem of a function f(x) is continuous on [a, b] and L is any…

Q: Prove that there are no rational numbers x and y such thatx2 - y2 =1002.

A: Consider the equation: x2-y2=1002. Find the rational numbers x and y satisfying the equation.…

Q: 9. Recall that Q denotes the set of all rational numbers and Z denotes the set of all integers.…

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