# 7 please
6. Prove that if x is rational and y is irrational, then 2 +y is irrational. 7. Prove that if x, y € R+ such that Ty # #4, then x + y.
5. Prove that v6 is not rational (it is irrational)
5. Prove that v6 is not rational (it is irrational)
(3) (a) Prove that, between any two rational numbers, there is an irrational number (b) Prove that, between any two irrational numbers, there is a rational number
Please help answer all parts!
(1) Prove that 75 is irrational. (State the Lemma that you will need in the proof. You do not need to prove the lemma.) (2) Disprove: The product of any rational number and any irrational number is irrational. (3) Fix the following statement so that it is true and prove it: The product of any rational number and any irrational number is irrational. (4) Prove that there is not a smallest real number greater than...
QUESTION 6 Prove by contraposition: "For all real numbers rifr is irrational, then is irrational. (Must use the method of contraposition). Which of the following options shows an accurate start of the proof. Proof. Letr be a real number such that r is irrational. Also, assume that r= where a, b are integers with b+0. b a Proof. Letr be a real number such that r2 where a, b are integers with b 0. b Proof. Letr be a real...
Question 5. Let x,y E R. Prove that if x and y are irrational, then at least one of 2 + y and c-y is irrational
Prove that 72 is irrational. State the question more precisely first.
Problem VI.(15 pts.) Suppose that is an irrational number. 1. Prove that j + cannot be a rational number 9 with gl < 2. 2. Can j + be a rational number whose absolute value is greater than 2? Why or why not?
Consider the following statement: 2 ^ (1/3) , the cube root of 2, is irrational. (a) First, prove that if n^3 is even, then n is even, where n is an integer. (b) Now, using a proof by contradiction, prove that 2^(1/3) , the cube root of 2, is irrational.
4. [5 Pts] Prove that the product of a non-zero rational number and an irrational number is irrational. Can you use a direct proof? Why or why not?