Prove that if a and b are real numbers with a < b, then there are infinitely many rational numbers in the interval (a, b).

Q: Prove

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Q: Let a,n∈N. Assume that n≥2. Show that if a^(1/n) is rational, then it is an integer.

A: The detailed solution is as follows below:

Q: Suppose there is a real number r with the property that r + 1/r is anodd integer. Prove that r is…

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

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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: 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: Let a,b,c, p be rational numbers such that p is not a perfect cube. If a + b p3 +cp = 0, then prove…

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Q: 17. Show that the set Q of all rational numbers is dense along the number line by showing that given…

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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: 6. Use the argument in the proof of Theorem 2.1.4 to show that there does not exist a rational…

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

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

Q: prove that for all rational numbers r and s, rs is rational

A: Given that r and s are rational numbers. That is r and s can be written in the form r=ab and…

Q: 3) Prove that there do not exist two rational numbers whose sum is irrational.

A: To prove: There do not exist any two rational numbers such that their sum is irrational.

Q: Prove that it is not true that the product of two irrational numbers is irrational.

A: To check:   Whether the product of two irrational numbers is irrational or not.

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: Every odd integer is the sum of three odd integers. There exist an irrational number c and a…

A: True  proof Let us assume  that x is even So x=2z for some constant z Therefore x.y=2zy Which…

Q: Prove that if a is a nonzero rational number and ab is irrational, then b is irrational.

A: Given that, a is a non-zero rational number and ab is irrational. We have to prove that b is…

Q: Show that the set of rational numbers Q is countably infinite, with |Q| = |N].

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Q: 3. Prove that the sum of any two rational numbers is a rational number

A: Consider ab,cd are rational numbers with a, b, c, d are integers. Now add these two numbers ab+cd…

Q: prove that for all positive integers therre exists two rational numbers such that there product is…

A: Let us consider two positive integers say 1 and 2. Now we find two rational numbers between them :…

Q: Prove the following using the specified technique: (a) Prove by contrapositive that for any two…

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Q: Show that for every real number ɛ 8 > 3ɛ. greater than zero, there exist a real number 8 such that

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Q: Prove that if x is rational and x = 0, then 1/x is rational.

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

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Q: x and y be two real numbers such that x + y is rational.

A: Given statement:   x and y be two real numbers such that x+y is rational. To Prove:  If x is…

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

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Q: Prove the following using a proof by contrapositive: For every pair of real numbers x and y, if x is…

A: According to the given information, it is required to prove the following using a proof by…

Q: Prove that for every positive integer n the number (1 – /2)" is not ratioanal.

A: Given: The given number is, (1-2)n where n is the positive integer.   To prove: The number (1-2)n is…

Q: 3. Prove or disprove: there exist rational numbers a and b such that a/b is irrational.

A: Suppose there exist rational numbers a and b such that a/b is irrational.

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

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Q: Prove that there exists no rational number x such that x2 = p, where p is a prime number (A prime…

A: We need to prove that there exists no rational number x such that , x2 = p  ,  where p is prime…

Q: Prove between any two distinct real numbers there is a rational number and an irrational number

A: Here we need to prove thatb any two distinct real numbers there is a rational number and an…

Q: 3. Suppose a is an irrational positive real number. Prove that if x and y are integers and x # y…

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Q: Prove or negate that there is a rational number x and an irrational number y such that x^y is…

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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: 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: Prove that the set of all positive rational numbers has no smallest element.

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Q: Prove that there are as many positive real numbers as there are real numbers.

A: Here , we have prove that all positive numbers are real numbers.

Q: Prove or disprove that if a and b are rational numbers, then ab is also rational.

A: we just give one example so that  a and b are rational but ab  is not rational

Q: Use the Intermediate Value Theorem to prove that for every real x > 0 and every integer n > 0, there…

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Q: Use the argument in the proof of Theorem 2.1.4 to show that there does not exist a rational number s…

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Q: Prove for every x>0   if x is irrational, then x1/2 is irrational

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Q: List the first 3 positive prime numbers d in Z so that the quadratic integers in Q[Vd] are precisely…

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Q: Prove by contrapositive that for any two real numbers, x and y, if x is rational and y is…

A: We prove it by contrapositive statement that for any two real numbers x and y, we will prove that if…