# Prove |A-B|=|A|: An Uncountable Set Solution

• ranykar
In summary, the conversation is about proving that the cardinality of a countable subset B of an uncountable set A is equal to the cardinality of A. The question is how to show a one-to-one correspondence between A and A-B. One person suggests using a proof by contradiction, while another suggests reordering A so that B is an initial segment.
ranykar
hello I am struggled with a qustion
let B be a countable subset of uncountable set A.
Prove |A-B|=|A|
i know how to prove that A-B is uncountable
but how do i show 1:1 with A?

Last edited:
welcome to pf!

hello ranykar! welcome to pf!

(no, you're struggling with the question!

or being strangled by it? )​

hint: assume the opposite

(ie, that A-B is countable)

tnx but that's not what i was asking.
i know that A-B is uncountable.
but how do i show 1:1 with A??

Try doing a reordering of A , so that B is an initial segment of A.

I think s/he wants to set up an actual bijection between A and A\B .

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

An uncountable set is a set that has an infinite number of elements, and those elements cannot be counted one by one. In other words, the elements of an uncountable set cannot be put into a one-to-one correspondence with the counting numbers (1, 2, 3, ...).

## 2. How is the absolute value of a set calculated?

The absolute value of a set is calculated by counting the number of elements in the set. For example, if a set contains 3 elements, its absolute value is 3. If a set is infinite, its absolute value is denoted by the symbol ∞ (infinity).

## 3. What does it mean for two sets to have equal absolute values?

When two sets have equal absolute values, it means that they have the same number of elements. In other words, they are considered to be the same size or cardinality.

## 4. How is the absolute value of the difference between two sets calculated?

The absolute value of the difference between two sets (|A-B|) is calculated by finding the number of elements that are in one set but not in the other. This can be done by subtracting the number of elements in the smaller set from the number of elements in the larger set. The result will always be a positive number.

## 5. How can it be proved that |A-B|=|A| for an uncountable set?

The proof for this statement relies on the concept of one-to-one correspondence. Since A is an uncountable set, it cannot be put into a one-to-one correspondence with the counting numbers. Therefore, any element that is removed from A (i.e. the set A-B) will still have the same number of elements as A. This means that |A-B| = |A|, as both sets have an uncountable number of elements.

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