Prove that if X is countable and a is in X then X \{a} is countable.

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In summary: That means that there exist some function f that maps natural numbers onto X. If a is in X, then there exist some natural number n so that f(n)= a. You an use that to show that X\{a} is countable.
  • #1
math25
25
0
Hi,

Can someone please help me with this problem.
Prove that if X is countable and a is in X then X \{a} is countable.

this is what I have so far, and I am not sure if its correct:

Every non-empty set of natural numbers has a least number. Since X is not a finite set, X must be an infinite set and thus X is nonempty in N. Suppose a in X is the least element. Now consider X\{a} Since X is infinite, X\{a} is an infinite subset of N. Then there is a least element a2 in X\{a}...

thanks
 
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  • #2
I would go about it this way: If X is countable, then there exists a subset of the natural numbers, say S, such that the cardinality of set S, call it r, is equal to the cardinality of set X. The set {a} has a cardinality of 1, so the cardinality of X\{a} is equal to r - 1 = n (some natural number, since we know X is nonempty). Then there exists a set S' such that its cardinality is equal to n, and therefore we have that X\{a} is countable.

Perhaps I have left out some important information; I am not as smart as some of the people on here. I am only trying to help. Hopefully someone can support or refute my ideas here.
 
  • #3
A subset of countable set is countable. Since
[tex]X\setminus\{a\}\subseteq X[/tex]

we have that [itex]X\setminus\{a\}[/itex] is countable. So try proving first that any subset of a countable set is countable
 
  • #4
so,is this correct

let x be countable set and x\{a} is subset of X

if x\{a} is finite it is countable

assume it is infinite then the function f : x|{a} ----> x defined by f(x0=x is 1-1 therefore x\{a} is countable


Also, can someone help me to prove or give a counter example is sum ai and sum bi are convergent series with non-negative terms then sum aibi converges
 
  • #5
Let A be a subset of a countable set B. By the definition of a countable set, there exists a one-to-one function f : B→N. Now restrict the domain of f to A to give a new function
g : A→N. g is one-to-one too because f is. So g is a one-to-one function that maps A into N. Thus A is countable

and start new thread for new problem
 
  • #6
thank you
 
  • #7
math25 said:
Hi,

Can someone please help me with this problem.
Prove that if X is countable and a is in X then X \{a} is countable.

this is what I have so far, and I am not sure if its correct:

Every non-empty set of natural numbers has a least number. Since X is not a finite set, X must be an infinite set and thus X is nonempty in N. Suppose a in X is the least element. Now consider X\{a} Since X is infinite, X\{a} is an infinite subset of N. Then there is a least element a2 in X\{a}...

thanks
You are assuming that X is a subset of N and that is not given. All you are given is that X is countable- the set of all integers is countable, the set of all rational numbers is countable, ... Of course, if X is countable, there exist a one-to-one correspondence between X and the natural numbers.
 

Related to Prove that if X is countable and a is in X then X \{a} is countable.

1. What does it mean for a set to be countable?

A set is countable if its elements can be placed into a one-to-one correspondence with the natural numbers (1, 2, 3, ...). This means that each element in the set can be assigned a unique number, and all elements in the set can be counted.

2. How can you prove that X \{a} is countable if X is countable and a is in X?

If X is countable, then we can arrange its elements in a sequence (x1, x2, x3, ...). Since a is an element of X, it must appear in this sequence. By removing a from this sequence, we are left with a new sequence (x1, x2, ..., xn-1, xn+1, ...). This new sequence is still a one-to-one correspondence with the natural numbers, thus proving that X \{a} is countable.

3. Can you provide an example to illustrate this proof?

Sure, let's say X = {1, 2, 3, 4, 5} and a = 3. We can arrange the elements in the set in a sequence as follows: (1, 2, 3, 4, 5). By removing 3 from this sequence, we are left with (1, 2, 4, 5), which is still a one-to-one correspondence with the natural numbers.

4. What is the significance of proving that X \{a} is countable?

This proof is significant because it shows that removing a single element from a countable set does not change its cardinality. In other words, if a set is countable, then removing one element from it does not make it uncountable.

5. Are there any other ways to prove that X \{a} is countable?

Yes, there are other ways to prove this statement. One way is to use the fact that a subset of a countable set is also countable. Since X is countable, X \{a} is a subset of X and therefore must also be countable. Another way is to use the definition of countability and show that X \{a} can be placed into a one-to-one correspondence with the natural numbers.

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