(a) Suppose x is an irrational number. Show that there is an integer n such that |x – n| < 1/2. (b) Prove or disprove that the product of two irrational numbers is always an irrational number.

Q: 5.44. Show that there exist two distinct irrational numbers a and b such that a" is rational. 5.45.…

A: We will show that there exist no real a, b such that 5.45 holds.

Q: 4. (a) Let Q= { a, b e Z, b+0, gcd(a, b) = 1 be the set of rational numbers. An integer p is called…

A: Assume, on the contrary, that √p is rational,so: √p = a/b for integers a and b, with the fraction…

Q: Consider the following theorem:If r and s are rational numbers, then the product of r and s is a…

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Q: (g) Prove that if x is a rational number and y is an irrational number, then x + y is an irrational…

A: Let x be a rational number and y is an irrational number . Claim: x+y is an irrational numbery  On…

Q: Prove that 2" < n! for any integern 2 4

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: 1. Prove that the square root of 2 is not a rational number.

A: Definition of rational number : If x is a rational number then it can be written in the form of pq…

Q: 6. Use a direct proof to show that the product of two odd numbers is odd. 8. Prove that if n is a…

A: We know that every odd numbers are of the type  2n-1 , where n is an integer.

Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

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

A: We have to prove that the product of two irrational numbers is not necessarily an irrational number.…

Q: Prove that √2 + √7 is an irrational number

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Q: 1. Prove that the negative of any even integer is even. 2. Prove that the difference between an even…

A: "Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: 14. Prove that if x is rational and x # 0, then 1/x is rational. 34. Show that these statements…

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Q: Prove the following by using: a contrapositive argument; the fact that a sum of integers is an…

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Q: Prove that for each irrational number r, there exists a unique integer n such that |r – n| < 1/2.

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Q: Q4. The decimal representation of some of the rational numbers may never end such as 1/3=…

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Q: Prove by contradiction: If x is a nonzero rational number and y is an irrational

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Q: 4. (a) Let Q = : a, b e Z, b#0, gcd(a, b) = 1} be the set of rational numbers. An integer p is…

A: Since we one(first) of the multiple question, so only question (a) will be answered.Please resubmit…

Q: 4. (a) Let Q = {: a, b € Z, b+ 0, ged(a, b) = 1} be the set of rational numbers. An integer p is…

A: Given: Q=ab: a,b∈z,b≠0,gcd(a,b)=1 and p is prime number.

Q: True or False Rational numbers have decimals that either terminate or are nonterminating with a…

A: True or False Rational numbers have decimals that either terminate or are nonterminating with a…

Q: Can you show me step by step how to do a proof by contradiction for the theorem that if the result…

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Q: Let { be an irrational real number. Let r be a non-0 rational number. Show that rg and r+{ are…

A: Given ξ be an irrational and r is a rational which is not equal to zero. r=pq where p and qare not…

Q: P Express 3.927927927. .. as a rational number, in the form where p and q have no common factors.…

A: Given:- 3.927927927......  To find:- Express given expression into p/q form

Q: 1. Prove that V3 is irrational number.

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Q: irrational

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Q: Show between any two irrational numbers there is an irrational number. (Hint: Assume r and s are…

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Q: "If r is a non zero rational number then r square root of 2 is an irrational number." For each…

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Q: _1. Which of the following statements is NOT TRUE? A. Every rational number is a real number. B.…

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Q: state the proof strategy you use

A: There are different ways of proving a result. It can be direct proof or indirect proof. In the case…

Q: Use proof by contradiction to show that 2 is not a rational number. Note that you may use the…

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Q: Do you think it is worthwhile, or useful, to solve the puzzles proposed in Euclid's theorem of…

A: The objective is to solve the puzzles proposed in Euclid's theorem of primes There are infinitely…

Q: Example 11 : Show that 32 is irrational. Solution : Let us assume, to the contrary, that 3-/2 is…

A: To prove: 32 is irrational   Suppose on the contrary, 32 is a rational number. Then by definition of…

Q: Prove by contradiction: The difference between any rational number and irrational number is…

A: Assume that x is an irrational number and y is rational number and assume that x – y is not…

Q: 8. (a) Prove that the sum of a rational and an irrational number is irrational. (b) Prove or…

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Q: A 16. Let x be a rational number and y be an irrational number. Which informal proof uses an…

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Q: All integers are rational

A: We know that "A number in the form of pq where (q≠0) is called as a rational number.

Q: Prove that √5 is irrational. (That is, prove there is no rational number x such that x2  = 5.)

A: The proof that 5 is an irrational number, can be done with the help of the contradictory assumption.…

Q: 6. Prove that if x is an irrational number and Y is a nonzero rational number, then xy is…

A: to prove this we take xy is rational number then show this is contradiction.

Q: Based on the above information, conjecture if the statement is ALWAYS true, SOMETIMES true, or NEVER…

A: Given- The product of a rational number and a rational number is rational. To find- The statement is…

Q: n fn(x) = D = 1+ x + k! +. . ·+ 2 6 n! k=Q

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Q: 4. (a) Let Q = : a, b e Z, b 0, gcd(a,b) = 1} be the set of rational numbers. An integer p is called…

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Q: For any natural number n, n²+1 is always odd. True False

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Q: a) Prove that if n is an even integer, then n2 + 5n+2 is even. (Use a direct proof.) b) Prove that…

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Q: 7a) Prove that if x is a rational number such that 2x-3 and 5x-4 are integers then x must be an…

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Q: The subject is DISCRETE MATH   Give a direct proof of the theorem “The sum of an even integer and…

A: Assume even and odd integers p and q respectively. Thus, p and q can be written as: p=2m and q=2n+1…

Q: (1). Prove that adding an irrational number with a rational number is an irrational number, and show…

A: Note:-Our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: (a) Prove that the real number V3 is an irrational number. That is, prove that If r is a positive…

A: Rational number: A number is known as rational number if it can be expressed as the ratio of two…

Q: The greatest integer in a real number a, denoted by x, is the largest integer less than or equal to…

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Q: 6. Prove that if the product of two real numbers is irrational, then at least one of the numbers is…

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