Problem B. (3 pts) Show that V-5 is not a rational mumber (i.e, irrational number).
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?
(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
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?
Proof by contradiction that the product of any nonzero rational number and any irrational number is irrational (Must use the method of contradiction). Which of the following options shows an accurate start of the proof. Proof. Let X+0 and y be two real numbers such that their product xy=- is a rational number where c, d are integers with d 0. Proof. Let x0 and y be two real numbers such that their product xy is an irrational number (that...
PROBLEM 3. Prove or disprove the following: /V2V54 log2 (J18 V ) is an irrational number. PROBLEM 4. Find the number of different symmetric relations that can be defined on a set 1 - {a,b). PROBLEM 5. Let A - {2, 3, 4, 8, 9, 12), and let the relation Ron A be defined by aRb if and only if (abia#b). Find R.
(b) A sequence (rn) of rational numbers having a limit lim r, that is an irrational number. Justify your examples using the definition of a limit.
Homework Problems Problem 1.10. The fact that that there are irrational numbers a, b such that a' is rational was proved in Problem 1.5. Unfortunately, that proof was non constructive it didn't reveal a specific pair, a, b, with this property. But in fact, it's easy to do this: let a ::= V2 and b::=2 log23. We know 2 is irrational, and obviously ab = 3. Finish the proof that this a, b pair works, by showing that 2 log...
pleases show all steps thankyou and write clearly Let a and b be rational numbers, and an irrational number. Let S be the statement t aH b then is irrational b+i ()Write down the converse and the contrapositive of S >p (i) Converse: & (ii) Contrapositive: a 77 bAX (b) Write down a proof that S is true.
Problem3 Prove each of the following statements below a. The product of a nonzero rational number and an irrational number is irrational. b. If n is an integer and n3 +5 is odd, then n is even.
code the inverse function 5. Here is the outline of a Rational number class, i.e., a class that contains fractional numbers represented by an integer numerator and denominator). class Rational ( << operator operator public: Rational (int num, int denom) operator inverse function moiionu o loino private: int num, denom For partial credit for parts a-c, you can code the *, Integer class and << operators for the a. Code the operator; the result of multiplying two rationals is a...