Proof: there is M s.t for all r>M, 1/2r < 1/100

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In summary: You can follow the flow from the conclusion and find out what needs to be true for it to be valid. It's a useful skill.In summary, the conversation is about expressing a statement using logical symbols and simplifying it to make it easier to understand. It also discusses the use of additional restrictions in front of the "for all" or "there exists" symbols, and the importance of being pedantic when learning about logical symbols and proof techniques.
  • #1
Aziza
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"Prove that there exists an odd integer M such that for all real numbers r larger than
M, [itex]\frac{1}{2r} < \frac{1}{100}[/itex]"

How to express this using logical symbols?

Literally I understand this statement as:
[itex](\exists M\in [/itex][PLAIN]http://upload.wikimedia.org/wikipedia/en/math/3/f/3/3f3c78f02a9c53f5460f4bcc2e7dd3cb.png[itex])[([/itex]M [Broken] is odd[itex])\wedge(\forall r\inℝ)(r>M\wedge\frac{1}{2r} < \frac{1}{100})][/itex]

But all r in ℝ cannot be greater than some M. There will always be r<=M if the only restriction on r is that it is in ℝ. So this statement can't be proven true. But would it be wrong to interpret the statement as follows:

[itex](\exists M\in [/itex][PLAIN]http://upload.wikimedia.org/wikipedia/en/math/3/f/3/3f3c78f02a9c53f5460f4bcc2e7dd3cb.png[itex])[([/itex]M [Broken] is odd[itex])\wedge(\forall r\inℝ)(r>M\Rightarrow\frac{1}{2r} < \frac{1}{100})][/itex]

(the last "and" changed to "imples")...then M=51 would prove the statement true...
 
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  • #2
Aziza said:
"Prove that there exists an odd integer M such that for all real numbers r larger than
M, [itex]\frac{1}{2r} < \frac{1}{100}[/itex]"

How to express this using logical symbols?

Literally I understand this statement as:
[itex](\exists M\in [/itex][PLAIN]http://upload.wikimedia.org/wikipedia/en/math/3/f/3/3f3c78f02a9c53f5460f4bcc2e7dd3cb.png[itex])[([/itex]M [Broken] is odd[itex])\wedge(\forall r\inℝ)(r>M\wedge\frac{1}{2r} < \frac{1}{100})][/itex]

[tex]\forall r \in \mathbb{R} > M \; ; \; \exists M = 2n-1 \; \wedge \; n \in \mathbb{Z} \; : \; \frac{1}{2r} < \frac{1}{100}[/tex]

But all r in ℝ cannot be greater than some M. There will always be r<=M if the only restriction on r is that it is in ℝ.
but that's not the only restriction on r. You wrote:

"Prove that there exists an odd integer M such that for all real numbers r larger than
M
..."

It's asserting that we can always find some odd integer M which makes any real number bigger than M satisfy the relation 1/2r < 1/100.

My only concern is that this looks a tad on the trivial side. Is the exercise just in writing and interpreting the symbols?
 
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  • #3
Simon Bridge said:
[tex]\forall r \in \mathbb{R} > M \; ; \; \exists M = 2n-1 \; \wedge \; n \in \mathbb{Z} \; : \; \frac{1}{2r} < \frac{1}{100}[/tex]

but that's not the only restriction on r. You wrote:

"Prove that there exists an odd integer M such that for all real numbers r larger than
M
..."

It's asserting that we can always find some odd integer M which makes any real number bigger than M satisfy the relation 1/2r < 1/100.

My only concern is that this looks a tad on the trivial side. Is the exercise just in writing and interpreting the symbols?

Ohh true true...I didn't know you could write something like [tex]\forall r \in \mathbb{R} > M[/tex]...is this a common notation? I mean, would a professor accept it? I am preparing for my transition to advanced math course this fall so this is a concern for me...my book always only uses one restriction in front of the "for all" or "there exists" symbol, so I thought that any additional restriction could be connected with an "and" to the rest of the statement...so to negate [itex]\forall r \in \mathbb{R} > M[/itex], it would just be:
[itex]\exists r \in \mathbb{R} > M[/itex] , right?

So the following statement
"For every positive real number x, there is a positive real number y less
than x with the property that for all positive real numbers z, yz ≥ z."
can be expressed as:

[itex](\forall x\in\mathbb{R^+})(\exists y\in\mathbb{R^+}<x)(\forall z\in\mathbb{R^+})(yz≥z)[/itex]
right?
and yea these proofs themselves are meant to be trivial, it's just about using basic proof techniques and knowing how to express statements using logical symbols
 
  • #4
OK since it is the point of this section, you'd better stick to being pedantic for now. I've never been penalized for that sort of shorthand and, if you skim a bunch of journals you'll find that convention gets abandoned in favor or readability.

Sadly, what gets emphasized seems to depend on the school.

What I wanted to show you was that you can make more sense out of things by working backwards.
 

What does the statement "Proof: there is M s.t for all r>M, 1/2r < 1/100" mean?

This statement means that there exists a number M, such that for all numbers larger than M, when divided by 2, the result is smaller than 1/100.

Why is this statement considered a proof?

This statement is considered a proof because it provides evidence or a logical argument to support the claim that for all numbers larger than M, when divided by 2, the result is smaller than 1/100.

How can this statement be applied in scientific research?

This statement can be applied in scientific research by providing a numerical limit that can be used to compare and analyze data. It can also be used to support or refute hypotheses related to numbers or numerical relationships.

Is this statement universally applicable to all numbers and situations?

No, this statement is only applicable to numbers and situations that involve division by 2 and the comparison of numerical values. It may not be applicable to other mathematical operations or concepts.

What is the significance of this statement in the scientific community?

This statement is significant in the scientific community because it showcases the use of numerical evidence and logical reasoning to support claims and hypotheses. It also highlights the importance of setting numerical limits and parameters in research and analysis.

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