14. Prove that if x is rational and x # 0, then 1/x is rational. 34. Show that these statements about the real number x are equivalent: (i) x is rational, (ii) x/2 is rational, (iii) 3x - 1 is rational. 2. Use a proof by cases to show that 10 is not the square of a positive integer. [Hint: Consider two cases: (i) 1sx< 3, (ii) x 2 4.]

Q: 9. We know that v2 is irrational. Use this fact and the previous exercise to prove that if r < s are…

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Q: 3. Prove that (v2+ v3) ¢ Q. You can use Prob. 2's result if needed. You may also use the fact,…

A: To prove 2+3∉ℚ That is 2+3 is irrational number Since the sum of two irrational numbers is…

Q: What is the largest root of the polynomial x? + x – 12? О А. 3 В. 4 О с. 1 O D. 2

A: Using factorisation to find roots

Q: Show that if x > 0, then 1 + 1x − x² ≤ √1+x≤ 1 + ½ x. -

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Q: Show that if x, y are rational numbers, then x + y and xy are rational numbers.

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Q: Definition. A real number x is rational if it may be written in the form x = 2 where p is an integer…

A: Given, that x is a rational number if it may be written in the form x=pq, where p is an integer and…

Q: 20. Prove: Let x and y be real numbers. If x is rational and y is irrational, then x + y is…

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Q: (c) Let us assume the facts from part (b) for all a, b e R, i.e. as < bs whenever a < b. Let SCR be…

A: Given that S~=x∈ℝ|x3∈S And S⊂ℝ is bounded above. We say that S⊂ℝ is bounded above if  and if…

Q: Show v2 is not rational and then conclude that the following set in Q does not have a supremum; A =…

A: We will show root 2 is irrational by first supposing that it is a rational and in the end showing…

Q: Which of the following is not a real number? (1) (6 + 2i)(6 – 2i) (2) (–2i)12 (3) (3 + 4i)(-4i) (4)…

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Q: (a) Recall that √√r> 0 for every positive real number r. Prove that if a and b are positive real…

A: are positive real number then we have to show

Q: 4. Show that Archimedean Principle is a consequence of the assertions that every interval (a, b)…

A: To prove that the Archimedean Principle is a consequence of the assertions that every interval a, b…

Q: We assume that x0. Therefore there exists a natural number n such that n> Since z is fixed we can…

A: Suppose that x and y are 2 distinct real numbers. We have to prove that, there exists a rational…

Q: 24 Selve the erosewond pele 0IRIDINATIE ENGINDE ET DEGREE C. Across: Down: 1. A number that can be…

A: We find the all above as shown below .just you need to put them in boxes because image is not clear

Q: Need

A: Given that, a can be written as base 10 x= a0+10a1+ ......+10nar We know that, If a divides b…

Q: 2. Which of the following are all the possible rational zeros for the polynomial x - 3x + 27x - 8?…

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

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Q: 5. For real numbers x and y, recall that max(x, y) is the bigger number between x andy and min(x, y)…

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Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: 7. Let y be irrational and x ‡ 0 a rational number. Show that the product xy is irrational.

A: We will prove given statement by method of contradiction.

Q: (b) Define x = a³ as the real number satisfying the equality (x · x)· x = a. For a > 0, you may…

A: As per  bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: (12) (a) Prove that if x € Z and r² is even, then x is even. (b) State and prove an analogous…

A: We prove following statements by using contrapositive method.

Q: Prove or disprove each of the following statements. To disprove just give a counter example. If n is…

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Q: 6. Prove that x is a rational number if and only if 2x – 1 is a rational number.

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Q: (i) Show that √6 + √7 is an algebraic number.

A: We are answering only the first question as per the guidelines. Please repost the other question to…