Consider the following statement: For all r E Q, s E R, rs is rational s is rational. (i) Which of "if", "only if", "iff" in make this a true statement? Prove your claim. (ii) Which of "if", "only if", "iff" in make this a false statement? Prove your claim.

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Q: Prove or disprove that if a, b E Q are rational numbers, then ab is also rational.

A: Statement: If a and b are rational number then ab is also rational.

Q: Prove

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A: I have given the answer in the next step. Hope you understand that

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A: This is basic algebra problem

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Q: 3. Show that for any rational number r and any X>0, , In x= rln x'

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Q: Prove that v7 is irrational. Prove that v7 + 5 is irrational too.

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A: Definition (Proof by contrapositive): Let P and Q be two statements. If P ⇒ Q, then not Q ⇒not P…

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A: a) Given: a∈ℚ\ 0, b∉ℚ To prove: ab∉ℚ b)  Given: x∈ℝ-ℚ, y∈ℝ-ℚ+  To prove: x+y∈ℝ\ℚ Rational Number: A…

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

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

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A: The given statement is false.

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

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Q: |Use the method of proof by contradiction to prove that 3 – V7 is irrational. (Hint: you can use the…

A: rational number  can be expressed as the quotient or fraction a/b of two integers.

Q: 3. Prove or disprove: there exist rational numbers a and b such that a/b is irrational.

A: Suppose there exist rational numbers a and b such that a/b is irrational.

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A: Solution c: Given that A=2112 We can write it as A=212×16⇒A=21216⇒A=216 Since 2 is a irrational…

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Q: Prove or disprove that if a and b are rational numbers, then ab is also rational.

A: we just give one example so that  a and b are rational but ab  is not rational

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Q: Give an indirect proof of the claim: Vr E R, if r³ is irrational then r is irrational.

A: There are two types of indirect proof :1)proof by contrapositive…