Prove that the set of irrational numbers is neither open nor closed.

Q: Prove that Va - Vb is irrational number if 'a' and 'b’ are prime number?

A: Given To prove that a-b is irrational number if a and b are prime number.

Q: Prove that the set of positive integers is equal to the set {1,2,3,...}. (Using Axioms)

A: This is a problem of set theory and cardinality.

Q: What is the set Z+ of all positive integers?

A: Set of integers are represented as: ℤ={.....,-3,-2,-1,0,1,2,3.........} Set of natural numbers:…

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: Outline a proof of the theorem by contradiction

A: On considering the theorem If A and B are comingled matrices then the product AB is commingled…

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: Prove that there is no rational number whose square is 12.

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Q: 2. The set of all rational numbers in the interval (0, 1).

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Q: If the rational number r is also an algebraic integer, prove that r must be an ordinary integer.

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Q: Is there a real number that is bigger than all integers? If so what is the number and if not what…

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Q: Prove that 3 is irrational number.

A: Rational Number: A number is of the form of pq, where p, q∈ℝ and gcd(p, q) = 1 Now we have to show…

Q: Give an example of an irrational point in Cantor Set.

A: There are infinitely many irrationals in cantor set ... one example is  ' an number that has base 3…

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: Is it possible for the set of rational numbers Q to be isomorphic to R? Explain.

A: Answer :  No , ℚ and ℝ are not isomorphic as rings. Explanation :              Suppose f:ℝ→ℚ be any…

Q: Prove that if a is an irrational number, then a-1 is an irrational number.

A: Given:           a is an irrational number To prove:            a-1 is an irrational number

Q: Show that the natural numbers and the rational numbers have the same cardinality.

A: Let integers Z are countable, we might wonder whether other infinite sets are also countable. We…

Q: Prove that there exists irrational numbers r and y such that x" is

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Q: Show that the set of irrational numbers is not a closed set.

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Q: Find a set of four integers such that their order and the order of their absolute values are the…

A: We have to find a set of four integers such that their order and the the order of their absolute…

Q: show that the is not complete. set of all rational numbers

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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: The product of an irrational number and a nonzero rational number is irrational.

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Q: Explain a) Why the set of rational numbers is countable. b) How to prove that a universally…

A: (a) Query- Why the set of rational numbers is countable. Answer- The set of all rationals in 0,1 is…

Q: Prove that the subset R/Q , the set of all irrational numbers , of R id of second category .

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Q: Prove that if sum of two numbers is irrational then at least one of them is irrational. Use indirect…

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Q: If the rational number r is also an algebraic integer, prove that r must be an ordinary integer.

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Q: 5. Prove that if x is an irrational number, then Vx is also irrational. (Domain: positive real…

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Q: Are some natural numbers  irrational

A: solution:-

Q: Prove by contradiction: V5 is an irrational number.

A: Let 5 is rational number ,  therefore, it will of the form pq  now 5=pq where p and q are both…

Q: Prove that the difference between two integers is odd if and only if exactly one of the integers is…

A: Suppose x be any integer. If x is odd then x=2m+1 for some integer m. If x is even then x=2m for…

Q: Does there exist a perfect set consisting of only rational numbers?

A: Given: Does there exist a perfect set consisting of only rational numbers?

Q: Prove that there does not exist an irrational number whose square root is rational.

A: Since, (p/q)=p^2/q^2 is rational. The square of any rational is rational, hence no rational is the…

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

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Q: Prove that for each real number x and each irrational number q, (x +q) is irrational or (x-q) is…

A: Note:  We will be solving Q. 4 only as asked.   In this solution we will prove that for any real…

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: What does the Subsets of set of real numbers means? And what does it consists of?

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Q: Use Proof by Contradiction to prove that the sum of an irrational number and a rational numbe is…

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Q: Prove that √7 is an irrational number.

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Q: Prove that, if z is an irrational number, then Va is also an irrational number.

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Q: What is the set Q of all rational numbers?

A: Set of integers: ℤ={.......-3,-2,-1,0,1,2,3,......} Set of natural numbers: ℕ={1,2,3......} Set of…

Q: 1. Prove that 3 is irrational number.

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Q: Use a proof by contradiction to show that the difference of any rational number and any irrational…

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Q: Prove that the set of irrational numbers in [0,1] is not countable.

A: The objective is to prove that the set of irrational numbers in 0,1 is not countable. Let's prove…

Q: Prove or Disprove: The set of all irrational numbers is uncountable.

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Q: What is the set Z of all integers?

A: The set of all integers is, ℤ=........, -4, -3, -2, -1, 0, 1, 2, 3, 4, ........