Simple question about induction .

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In summary, the conversation is about proving the functional equation f(x)=x^2 for a natural number x. The speaker suggests using induction to prove this equation, and also mentions the importance of knowing the value of f(x-1). The conversation ends with the speaker stating that they have solved the issue and everything is clear now.
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Andrax
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This is from a functional equation , i think there is no points in posting it here this is what i ant to prove the functions is from N to N .

let x be a natural number
i want to prove that f(x)=[tex]x^2[/tex]

By induction
suppose that f(x)=[tex]x^2[/tex]
f(0)=0 holds (from the functional equation)

i want to prove that f(x)= [tex](x+1)^2[/tex]

when putting (x,1)(we have f(x+y)+f(x-y) in the equation) I need to know the value of f(x-1) ,does the assumption that f(x)=[tex]x^2[/tex], imply that f(x-1)=[tex](x-1)^2[/tex]
(this is not a homework i just saw this excercise online and decided to do it).
 
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  • #2
Where is the functional equation?
Do you have a question, or do you want to start a discussion (about what?)?

If you want to prove f(x)=x2, you cannot just assume that this is true (without additional statement, this means "for all (natural) x"). It ruins the idea to prove it.
 
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i solved the issue now , thanks ,everything is clear,
( i meant for an unspecific x let f(x))=x^2 then i'd have to prove by induction ..
 

What is induction?

Induction is the process of reasoning in which a conclusion is drawn based on a pattern of observed evidence or events.

What is the difference between deductive and inductive reasoning?

Deductive reasoning starts with a general principle or theory and applies it to a specific case, while inductive reasoning starts with specific observations and uses them to make a general conclusion.

How is induction used in science?

Induction is used in science to make generalizations and predictions based on observed patterns and evidence. It is a crucial tool in developing scientific theories and hypotheses.

What are the limitations of induction?

One limitation of induction is that it relies on the assumption that past patterns will continue to hold true in the future. It is also vulnerable to bias and can lead to false conclusions if the evidence is not representative or if there are hidden variables at play.

Can induction be used to prove a hypothesis?

No, induction cannot prove a hypothesis to be true. It can only provide evidence to support a hypothesis, but it cannot confirm it with certainty. Scientific theories are always open to revision and further testing.

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