Answered: Prove the following statement using a…

Prove the following statement using a direct proof. If x and y are rational numbers, then 4x + y is also a rational number.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter7: Real And Complex Numbers

Section7.1: The Field Of Real Numbers

Problem 8TFE: Label each of the following statements as either true or false. Every decimal representation of a...

See similar textbooks

Related questions

Q: Prove the following using the specified technique: (a) Prove by contrapositive that for any two real…

Q: Decide whether the following statement is true or false, and prove your claim: For all a∈R,a is…

Q: Prove the following using a proof by contrapositive: Let z be a rational number. Prove that if ry is…

Q: Consider the following statement. If r is any rational number and s is any irrational number, then…

Q: If a and b are integers, then a/b is always a rational number. True False

Q: Which of the following statements are true of irrational numbers?

Q: If æ is an irrational number, then it is the case that sup{q E Q : q< x} < x.

Q: Prove

Q: Use proof by contradiction to prove the following statements. 1. The sum of a rational number and an…

Q: 1. If x is an odd integer, then x3 is odd.

Q: Prove the following using the specified technique: (a) Let x and y be two real numbers such that r…

A: For part a: The objective here is to prove that if x is irrational, then x-y is irrational. Given: x…

Q: Prove the following using the specified technique: Let x and y be two real numbers such that x+y is…

A: Consider the given information. Given that x and y be two real numbers such that x+y is rational. To…

Q: 3. Write the following as symbolic statements. (a) There exists a rational number q such that qr = r…

Q: Prove the statement if it is true, and give a counterexample if it is not true. a. For every real…

A: Both the subparts are solved with explanation.

Q: Use direct proof to prove the following statements. 1. Suppose a,beZ. If alb, then al(3b – b² +5b).…

A: This is basic algebra problem

Q: Prove the following statements using direct and indirect proof methods. a.) For all integers a, b,…

A: a.) For all integers a, b, c, the product (a−b)(a−c)(b−c) is even. The differences of any two even…

Q: 1) Prove by contradiction: a) If x is a rational number, then x + V2 is an irrational number. b) If…

A: Given a) If x is a rational number To prove that is an irrational number.

Q: If the rational number r is also an algebraic integer, prove that r must be an ordinary integer.

Q: Prove the following statement: 'For every irrational number r and rational number s # 0, the product…

A: We have to prove the following statement- For every irrational number 'r' and rational number 's' ≠…

Q: Consider the following statement. If r is any rational number and s is any irrational number, then…

A: To find the a,b

Q: Analyze the following "proof" and determine if there is anything incorrect and why. Claim. Let x be…

A: We have to show that if x is irrational number then 2x is also irrational number

Q: Given: "if x is rational and x + 0, then 1/x is rational", can be proven by Direct proof. O True…

Q: prove that for all rational numbers r and s, rs is rational

A: Given that r and s are rational numbers. That is r and s can be written in the form r=ab and…

Q: Let x. yER. If *<y with x and y very close to each other. Then which of the following statements is…

A: Statement:- Let x, y∈ℝ. If x < y with x and y very close to each other.

Q: Decide whether the following statements are true or false.Provide counterexamples for those that are…

A: Every nonempty open set contains a rational number. This is a true statement.

Q: If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?

A: Since, R consists of real numbers which include both rational and irrational numbers.

Q: Given: "if x is rational and x + 0, then 1/x is rational", can be proven by Direct proof. False O…

Q: ) For each of the following statements, determine whether it is true or false. If you answer (P1,…

Q: Prove the following statement: "The sum of any two rational numbers is rational." 99

A: Concept Used: A rational number is the number that can be written in the form ab where , a and b are…

Q: Prove the following statement by contradiction. If a and b are rational numbers, b 0, and r is an…

Q: Prove the following using the specified technique: (a) Prove by contrapositive that for any two…

Q: 2 We know that & is clased under "+" and Moreover,we know that ヒnaw +hut B ER-8 and (5 ER-Q. Prove…

A: Given that3∈R-Q and 5∈R-Q Have to prove 3+5∈R is not a rational number.

Q: If S is a nonempty subset of the negative rational numbers, then: 3x e S(Vy e S(x > y)). O true O…

Q: Are the following statements true or false? Justify each conclusion. (c) For all integers a and h,…

A: c For all integers a and b, if a is even and b is odd, then 4 does not divide (a2 + 2b2 ).The…

Q: 1.* For each of the following statements, decide whether it is true or false. If true, give a proof.…

Q: Give an example of each of the following:(c) A rational number that is not an integer

A: Any number which can be represented as pq, where q≠0, is called a rational number. Ex. 23,010,51…

Q: is the completion of CX sd) is the set of all rational number

A: We know that a metric space (X,d) is complete iff every Cauchy sequence in X is convergent in X but…

Q: Prove by contradiction. If a and b are rational numbers, b ≠ 0, and r is an irrational number, then…

A: Assume that the number a + b r is rational. Since a and b are rational, then a and b can be…

Q: 6. Are the following statements true or false? Justify each conclusion. * (a) For each positive real…

A: Answer(a): The given statement is false. Because for each positive real number x, if x is irrational…

Q: x and y be two real numbers such that x + y is rational.

A: Given statement: x and y be two real numbers such that x+y is rational. To Prove: If x is…

Q: If S is a nonempty subset of the negative rational numbers, then: Jx e S(Vy E S(x > y)). true O…

A: Note: Rational numbers are countably infinite. So there is no largest rational number and no…

Q: Are the following statements true or false? Justify each conclusion. ' (a) For each positive real…

A: Hey, since there are multiple sub-parts posted, we will answer the first 3 sub-parts. If you want…

Q: Write a detailed proof for the statement: Between every two rational numbers there exists an…

Q: Verify that the embedding of ℚ ≥0 in ℝ ≥0 respects ≤. That is, verify for a, b ∈ ℚ ≥0 that if a ≤ b…

A: As the ordering ≤ is not given specifically for embedding of ℚ in ℝ, we will use the Dedekind cuts…

Q: Prove the following using a proof by contrapositive: Let a be a rational number. Prove that if ry is…

Q: for all integers a, b, and c, if a divides b and a divides c, then a divides (b2-c2). prove this…

A: Introduction: Divisibility is the base of number theory. For two integers a and b, if there exists…

Q: Which of the following statements is NOT true? A The set of natural numbers is a subset of integers.…

A: All the positive numbers 1, 2,3,,..... are called natural numbers.

Q: 1. Restate each supposition using the appre (a) Suppose r ERand (3z] 6y+5. By definition, %D (b) Let…

A: a) Given supposition is: 3x=6y+5

Q: Use proof by contradiction to prove the following statement: For all real numbers x and y, if x/y is…

A: To prove the given statement using contradiction, we will first assume that x is rational.

Question

Prove the following statement using a direct proof.
If x and y are rational numbers, then 4x + y² is also a rational number.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

Prove that if a is rational and b is irrational, then a+b is irrational.
If x and y are positive rational numbers, prove that there exists a positive integer n such that nxy. This property is called the Archimedean Property of the rational numbers. (Hint: Write x=a/b and y=c/d with each of a,b,c,d+.)
Prove by induction that if r is a real number where r1, then 1+r+r2++rn=1-rn+11-r
Prove that if is an irrational number, then is an irrational number.
13. Prove that if and are rational numbers such that then there exists a rational number such that . (This means that between any two distinct rational numbers there is another rational number.)
Use the fact that 3 is a prime to prove that there do not exist nonzero integers a and b such that a2=3b2. Explain how this proves that 3 is not a rational number.
Label each of the following statements as either true or false. Every decimal representation of a real number that terminates represents a rational number.
Show that if the statement 1+2+3+...+n=n(n+1)2+2 is assumed to be true for n=k, the same equation can be proved to be true for n=k+1. Explain why this does not prove that the statement is true for all positive integers. Is the statement true for all positive integers? Why?