Can infinite sets differ finitely?

In summary, the conversation discusses whether two infinite sets can differ by a finite number of elements, with examples given and clarification provided on the use of set subtraction and union. The concept of "is contained in" is also briefly mentioned as a potential source of confusion.
  • #1
Loren Booda
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4
Can two infinite sets differ by a finite number of elements?
 
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  • #2
Take for example [itex]A = \mathbb{N}[/itex] and [itex]B = \mathbb{N} \setminus \left\{ 1 \right\}[/itex]. Then [itex]A[/itex] and [itex]B[/itex] are both infinite sets, and they share all elements except for 1. Or is this not what you mean by "differ"?
 
  • #3
Loren Booda said:
Can two infinite sets differ by a finite number of elements?

I take your question as:

"Given two infinite sets A and B, is the symmetric difference [itex](A\cup B)\setminus(A\cap B)[/itex] ever nonempty?"

in which case the above post gives an example of when this can occur.
 
  • #4
Please forgive my ignorance, but what does \ mean in this context?
 
  • #5
Let' say A = {0, 1, 2, ...} and B = {1, 2, 3, ...}. A and B are both infinite, but A and B "differ" by one element. Namely 0.
 
  • #6
Loren Booda said:
Please forgive my ignorance, but what does \ mean in this context?

Set subtraction (thus the LaTeX command \setminus). {1, 2, 3} \ {2} = {1, 3}.
 
  • #7
CRGreathouse said:
I take your question as:

"Given two infinite sets A and B, is the symmetric difference [itex](A\cup B)\setminus(A\cap B)[/itex] ever nonempty?"

in which case the above post gives an example of when this can occur.

Loren Booda said:
Please forgive my ignorance, but what does \ mean in this context?
It's the "set difference". A\B is "All values that are in A but not in B". Think "A take away any members of A intersect B".
 
  • #8
Thanks, all.
 
  • #9
There isn't an inverse function to \, is there?
 
  • #10
Yes there is, the union. A \ B U B = A.
 
  • #11
Not quite; only if B is contained in A.
 
  • #12
secretman said:
Not quite; only if B is contained in A.

Only if B is a subset of A.

Using "is contained in" to mean "is a subset of" can cause an awful lot of confusion.
 

1. Can infinite sets have the same cardinality as finite sets?

Yes, it is possible for an infinite set to have the same cardinality as a finite set. This is because cardinality refers to the size or number of elements in a set, and not all infinite sets are larger than finite sets in terms of cardinality. For example, the set of all positive integers and the set of all even positive integers have the same cardinality, even though one is infinite and the other is finite.

2. How can infinite sets differ finitely?

Infinite sets can differ finitely if they have different cardinalities. This means that one set has a larger or smaller number of elements than the other, but both are still infinite sets. For example, the set of all positive integers and the set of all real numbers have different cardinalities, even though both are infinite sets.

3. Are there different levels of infinity in infinite sets?

Yes, there are different levels of infinity in infinite sets. This is because not all infinite sets have the same cardinality. For example, the set of all natural numbers is smaller than the set of all real numbers, even though both are infinite sets. This concept is known as the "infinity of infinities."

4. Can there be a one-to-one correspondence between two infinite sets?

Yes, it is possible for there to be a one-to-one correspondence between two infinite sets. This means that each element in one set can be matched with exactly one element in the other set. For example, there is a one-to-one correspondence between the set of all positive integers and the set of all even positive integers.

5. How do we determine the cardinality of infinite sets?

The cardinality of infinite sets can be determined by using the concept of bijection. A bijection is a function that maps each element in one set to a unique element in the other set. If there exists a bijection between two sets, then they have the same cardinality. However, if there is no bijection between two sets, then they have different cardinalities.

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