What is the flaw in the statement about T and how can it be corrected?

  • Thread starter Jamin2112
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In summary, the statement given contains a flaw as the set T and the set {n \in N : n ≥ m} are not necessarily equivalent. The correct statement would be, "If T is a set of natural numbers such that 1) m \in T and 2) n \in T implies n+1 \in T, then T = {n \in N : n ≥ m-1}." This ensures that all elements in T are greater than or equal to m-1, rather than m, which may not necessarily be in T.
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Jamin2112
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Homework Statement



Let m be a natural number. Find the flaw in the statement below. Explain why the statement is not valid, and change one symbol to correct it.

"If T is a set of natural numbers such that 1) m [tex]\in[/tex] T and 2) n [tex]\in[/tex] T implies n+1 [tex]\in[/tex] T, then T = {n [tex]\in[/tex] N : n ≥ m}

Homework Equations



Dunno.

The Attempt at a Solution



Part 2) of the if statement tells us that T is an infinite set. I'm not sure exactly how 1) and 2) are connected. Hmmmm ...

Help me get started.
 
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  • #2
To get started think about this. Is m-1 in T?
 
  • #3
Dick said:
To get started think about this. Is m-1 in T?

Hmmm ...

T is going to look something like {k, k+1, k+2, ...}, where k≥1 is an integer. That's basically what the second condition tells me.

m is some element in T. That's all I know about m. Could m-1 be in T? As long as m>k.
 
  • #4
So is their equation for T correct?
 
  • #5
Jamin2112 said:
Hmmm ...

T is going to look something like {k, k+1, k+2, ...}, where k≥1 is an integer. That's basically what the second condition tells me.

m is some element in T. That's all I know about m. Could m-1 be in T? As long as m>k.

Ok, so you don't know if m-1 is in T. On the other hand, m-1 is definitely NOT in [m,infinity). That suggests that T and [m,infinity) are not necessarily the same thing.
 

1. What does "Let m be a natural number" mean?

"Let m be a natural number" is a common notation used in mathematics to introduce a variable, where m represents a positive integer. This notation is often used to define a problem or state a hypothesis.

2. What is a natural number?

A natural number is a positive integer that is used for counting and ordering objects. It is denoted by the symbol N and includes all whole numbers from 1 upwards.

3. Why is it important to specify that m is a natural number?

Specifying that m is a natural number is important because it restricts the possible values of m to positive integers only. This can help to narrow down the scope of a problem or hypothesis and make it more manageable to work with.

4. Can m be any positive number, or does it have to be an integer?

In mathematics, the natural numbers are defined as positive integers, so m must be an integer. This is because natural numbers are used for counting and ordering, and it would not make sense to have a non-integer value for m in this context.

5. How is "Let m be a natural number" used in mathematical proofs?

In mathematical proofs, "Let m be a natural number" is often used as a starting point to establish a hypothesis or to define a variable in an equation. It helps to clearly state the assumptions and conditions of the problem being solved, making the proof more logical and organized.

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