Proving Equivalence of Rational Number Statements

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In summary, a proof in science is a logical and systematic demonstration that provides evidence for a hypothesis or theory. It is important to have a proof in science in order to validate hypotheses and theories and make informed decisions about scientific concepts and discoveries. The key components of a proof include a clear hypothesis or claim, a systematic approach to testing, use of empirical evidence and experimentation, and a detailed explanation of results and conclusions. A proof is considered valid if it follows a logical and systematic approach, uses reliable and relevant evidence, and supports the initial hypothesis or claim. Some common mistakes to avoid when creating a proof include using biased or unreliable evidence, drawing unsupported conclusions, and failing to consider alternative explanations.
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Show that these statement about real number x are equivalent
(i) x is rational
(ii) x/2 is rational
(iii)3x-1is rational

i think you have to use contradiction but i don't know how
 
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x is rational if and only if you can write it in the form p/q for p, q integers (and q non-zero). So if x is rational, let x=p/q. Can you find integers m and n such that x/2=m/n? You can do this exact process to show that 1 implies 2, 2 implies 3, and 3 implies 1 (which gives that they are equivalent)
 

FAQ: Proving Equivalence of Rational Number Statements

What is a proof in science?

A proof in science is a logical and systematic demonstration that provides evidence for a hypothesis or theory. It involves using empirical evidence, experimentation, and logical reasoning to support a claim or idea.

Why is it important to have a proof in science?

Having a proof in science is important because it allows for the validation of hypotheses and theories. It provides a basis for understanding the natural world and making informed decisions about scientific concepts and discoveries.

What are the key components of a proof?

The key components of a proof include a clear hypothesis or claim, a systematic and logical approach to testing the hypothesis, use of empirical evidence and experimentation, and a detailed explanation of the results and conclusions.

How do you determine if a proof is valid?

A proof is considered valid if it follows a logical and systematic approach, uses reliable and relevant evidence, and is able to support the initial hypothesis or claim. It should also be open to scrutiny and able to withstand criticism.

What are some common mistakes to avoid when creating a proof?

Some common mistakes to avoid when creating a proof include using biased or unreliable evidence, drawing conclusions that go beyond the scope of the evidence, and failing to consider alternative explanations or interpretations. It is also important to clearly explain each step of the proof and avoid logical fallacies.

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