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The difference of any two irrational numbers is irrational.

The difference of any two irrational numbers is irrational.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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How do you write the proof for number 8?
Determine which statements in 3-13 are true and which are false. Prove those that are true and disprove those that are false. 4. 3√2-7 is irrational. 6. √2/6 is rational. 3.6-7√2 is irrational.com 5. √4 is irrational. 7. The sum of any two irrational numbers is irrational. 8. The difference of any two irrational numbers is irrational. 9. The positive square root of a positive irrational number is irrational. GES 10. If r is any rational number and s is any irrational number, then r/s is irrational. 11. The sum of any two positive irrational numbers is irrational. 12. The product of any two irrational numbers is irrational. H 13. If an integer greater than 1 is a perfect square, then its cube. root is irrational. 14. Consider the following sentence: If x is rational then √√x is irrational. Is this sentence always true, sometimes true and sometimes false, or always false? Justify your answer. Tom Toeb 4.6 Indirect A 15. a. Prove that for all integers a, if a³ is even then a is even. b. Prove that 2 is irrational. 16. a. Use proof by contradiction to show that for any inte- ger n, it is impossible for n to equal both 3q₁ +r₁ and 392 +r2, where 91, 92, 71, and r2, are integers, 0≤r₁ 3,0 ≤r₂ < 3, and r₁ r2. b. Use proof by contradiction, the quotient-remainder theorem, division into cases, and the result of part (a) to prove that for all integers n, if n² is divisible by 3 then n is divisible by 3. c. Prove that √3 is irrational. Se 17 Ci to deduce a contra di approach. nod H 23. Prove that √2 + v *24. Prove that log, (2) torisation of integ how that if d is not prime and n² is 22. Use the proof techn that if n is any pos then √n is irration: H 25. Let N = 2.3.5.7 N is divided by answer. H 26. Suppose a is an pla and pl(a+ 27. Let P1, P2, P3, ing order. Here i P1 2 P H a. For each i Calculate N b. For each i number q s primality f your answe For exercises 28 a 28. An alternativ begins as fol" Proof: Supp bers. Then o
Transcribed Image Text:Determine which statements in 3-13 are true and which are false. Prove those that are true and disprove those that are false. 4. 3√2-7 is irrational. 6. √2/6 is rational. 3.6-7√2 is irrational.com 5. √4 is irrational. 7. The sum of any two irrational numbers is irrational. 8. The difference of any two irrational numbers is irrational. 9. The positive square root of a positive irrational number is irrational. GES 10. If r is any rational number and s is any irrational number, then r/s is irrational. 11. The sum of any two positive irrational numbers is irrational. 12. The product of any two irrational numbers is irrational. H 13. If an integer greater than 1 is a perfect square, then its cube. root is irrational. 14. Consider the following sentence: If x is rational then √√x is irrational. Is this sentence always true, sometimes true and sometimes false, or always false? Justify your answer. Tom Toeb 4.6 Indirect A 15. a. Prove that for all integers a, if a³ is even then a is even. b. Prove that 2 is irrational. 16. a. Use proof by contradiction to show that for any inte- ger n, it is impossible for n to equal both 3q₁ +r₁ and 392 +r2, where 91, 92, 71, and r2, are integers, 0≤r₁ 3,0 ≤r₂ < 3, and r₁ r2. b. Use proof by contradiction, the quotient-remainder theorem, division into cases, and the result of part (a) to prove that for all integers n, if n² is divisible by 3 then n is divisible by 3. c. Prove that √3 is irrational. Se 17 Ci to deduce a contra di approach. nod H 23. Prove that √2 + v *24. Prove that log, (2) torisation of integ how that if d is not prime and n² is 22. Use the proof techn that if n is any pos then √n is irration: H 25. Let N = 2.3.5.7 N is divided by answer. H 26. Suppose a is an pla and pl(a+ 27. Let P1, P2, P3, ing order. Here i P1 2 P H a. For each i Calculate N b. For each i number q s primality f your answe For exercises 28 a 28. An alternativ begins as fol" Proof: Supp bers. Then o
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