Is it true for all n, Natural number?

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In summary, Matt Grime said that one may construct a statement P(n) that is true for odd n but false for even n.
  • #1
Willy_Will
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Hello all,

Another quick question for the number theory gurus here:

Let P(n) predicate, n Natural number. Suppose that P(n) satisfies that P(1) is true, and if k in N, P(k) is true, then P(k+2) is true. Is P(n) true for ALL n in N? Why?

Thanks in advace guys!

-William
 
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  • #2
The principal of induction sezs that if its true for N, it's true for N+1, and we begin with a basis of 1 or zero, usually. This means once we have proved it true for 5, we can deduce that it is true for 6, and if true for 6, well...

So if its true for N+2 when its true for N...you can take it from there.
 
  • #3
I would say no. P(1) -> P(3) -> P(5) and so on. In other words, P(n) is true for all odd n.
 
  • #4
Interesting e(hoOn3... I was thinking something like that, other opinions?
 
  • #5
No. The answer is no.

Well, depending on how N was constructed. You could construct N differently and then be able to say yes.
 
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  • #6
If you say no, why no? N = Natural numbers,

thanks for your reply!
 
  • #7
just construct a counter example. it is trivial. P(1), and hence P(3), P(5),.. are true, but P(2) etc can be completely arbitrary. Are you telling me you can't think of an obvious statement that is true for (something to do with) odd numbers but is false for even numbers?
 
  • #8
Glad there is edit here... its late, I am not thinking straight.
I didnt catch the counter example, sorry.
 
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  • #9
I am not following you at all.

Matt Grime pretty much gave you a counter example.
 
  • #10
Wrong. Matt didn't give a counter-example per say. He said that one may construct a statement P(n) that is true for odd n but false for even n.
 
  • #11
I think I pretty much did give a counter example. At least if you take second to think what the words ODD and EVEN are doing there. Isn't there a really obvious property that ODD numbers have that EVEN ones don't? And can you show it by induction? Yes. It is a vacuous proof, but still a proof by induction nonetheless.
 
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1. Is it true for all n that every natural number has a successor?

Yes, it is true for all n that every natural number has a successor. This is known as the successor function and is a fundamental property of natural numbers in mathematics.

2. Is it true for all n that every natural number is greater than zero?

Yes, it is true for all n that every natural number is greater than zero. This is because natural numbers are defined as positive whole numbers, and zero is not considered a natural number.

3. Is it true for all n that every natural number is less than its successor?

Yes, it is true for all n that every natural number is less than its successor. This is a property of the natural numbers known as the order property, which states that each natural number is greater than the previous one.

4. Is it true for all n that every natural number is either prime or composite?

No, this statement is not true for all n. While all natural numbers are either prime or composite, there are infinitely many natural numbers that are neither prime nor composite, such as 1, 2, and 3.

5. Is it true for all n that every natural number can be written as the sum of two prime numbers?

No, this statement is not true for all n. While every natural number can be written as the sum of two prime numbers (known as Goldbach's conjecture), it has not been proven to be true for all natural numbers. In fact, this conjecture is still an open problem in mathematics.

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