Proving A is Not Finite: B is Not Finite

  • Thread starter Ka Yan
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In summary, Ka Yan's classmate showed him another way to prove that A is infinite, by proving that there exists a function g on A, such that g(x)=f(x) for all x in B. This function is a bijection, so A is infinite.
  • #1
Ka Yan
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There is a simple problem, and I gave my simple prove. Could anybody help me check whether it is correct:

Show that if B is not finite and [tex]{B}\subset{A}[/tex], then A is not finite.

My prove:
Since B is not finite, there exists a bijection of B into one of its proper subset, C, say, and denote the function to be f: [tex]{B}\rightarrow{C}[/tex].
Since B is a proper subset of A, as an extension of f to A,there exist a (some) bijiection(s), say g, such that g: [tex]{A}\rightarrow{C}[/tex]. And thus g maps A onto its proper subset, A cannot be finite. Hense A is infinite.

Thks!
 
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  • #2
What, precisely, do you mean by "an extension of f to A". In other words, what, exactly, is g?
 
  • #3
You should also notice that there doesn't generally exist a bijection of A to C. Which is ok, because to show A is infinite you don't need a bijection with C, you only need one with 'some subset' of A.
 
  • #4
can't you simply say if [tex]{B}\subset{A}[/tex]

n [tex]\le[/tex] cardinality of (B) [tex]\leq[/tex] cardinality of (A) for all n in N.

So A is not finite.
 
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  • #5
fakrudeen said:
can't you simply say if [tex]{B}\subset{A}[/tex]

n [tex]\le[/tex] cardinality of (B) [tex]\leq[/tex] cardinality of (A) for all n in N.

So A is not finite.

Yes, if you have cardinality theorems available. Ka Yan appears to want to prove it using the definition that a set is infinite if there exists a bijection with a proper subset of itself.
 
  • #6
Yes, thanks gentlemen.

That, I was originally considered, if f is a function on a subset B of A, then an "extension" of f from B to A is that, exist a function g on A, such that g(x)=f(x) for all x in B.

But B and A are seem necessary to be closed, and I didnt quite sure if those A and B are closed or not to asure the concept "extension" be apply.

Besides, my classmate gave me another way:
construct a sequense {an} by: take a0[tex]\in[/tex]B, a1 [tex]\in[/tex] B-{a0}, a2 [tex]\in[/tex] B-{a0, a1}, and so on. And this sequense is infinite, and all belong to A, thus A is infinite.
I don't quite well remember the constructing process, and I showed just what he meant.
 
  • #7
Here's another way. Consider the subset D=(A-B) union C. It's a proper subset of A since C is a proper subset of B. Define g(x)=f(x) for x in B and g(x)=x for x in A-B. g is a bijection of A with a proper subset D. So A is infinite. I think that's the 'extension' you were after.
 
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What does it mean to prove that A is not finite and B is not finite?

Proving that A is not finite and B is not finite means demonstrating that both sets A and B have an infinite number of elements. This can be done through mathematical or logical reasoning.

Why is it important to prove that A is not finite and B is not finite?

Proving that A is not finite and B is not finite is important because it can help us understand the properties of infinite sets and their relationship to finite sets. It can also have implications in various fields such as computer science, physics, and philosophy.

How can one prove that A is not finite and B is not finite?

There are different ways to prove that A is not finite and B is not finite, depending on the context and the specific sets involved. One common method is to use a proof by contradiction, where we assume that the sets are finite and then show that this leads to a contradiction. Another method is to use mathematical induction, where we show that the sets follow a pattern that continues infinitely.

What are some examples of infinite sets?

Some examples of infinite sets include the set of natural numbers (1, 2, 3, ...), the set of real numbers (all numbers on the number line), and the set of positive even integers (2, 4, 6, ...). These sets can be proven to be infinite by demonstrating that they have a one-to-one correspondence with a larger infinite set.

Can A be finite while B is not finite?

Yes, it is possible for A to be finite while B is not finite. This means that A has a limited number of elements, while B has an infinite number of elements. This can be seen in the example of the set of natural numbers and the set of real numbers, where the former is finite while the latter is infinite.

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