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, ...).
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).
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.
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.
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.