Is an uncountable set minus a countable set countable?
We can see that an uncountable set minus a countable set is indeed uncountable. The union of countably many countable sets is countable; thus (A−B)∪B is countable. But then A is a subset of (A−B)∪B and thus must be countable itself, which is a contradiction.
How do you prove a set is countable or uncountable?
In mathematics, a set is countable if it has the same cardinality (the number of elements of the set) as some subset of the set of natural numbers N = {0, 1, 2, 3.}.
Do two uncountable sets have the same cardinality?
If a set has a subset that is uncountable, then the entire set must be uncountable. These sets are both uncountable (in fact, they have the same cardinality, which is also the cardinality of R, and R has infinite length). So by rearranging an uncountable set of numbers you can obtain a set of any length what so ever!
Is a is an uncountable set and B is a countable set must AB be uncountable?
However, finding a listing means that A is countable, but we assumed A was uncountable. Therefore, yes, A – B must be uncountable.
How do you prove an infinite set is uncountable?
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.
Does every uncountable set have an uncountable subset?
Every uncountable set admits uncountably many uncountable subsets. Every uncountable set admits uncountably many countable subsets. Every uncountable set admits uncountably many finite subsets. More generally, if , and , then admits a subset of cardinality .
How do you prove a closed interval is uncountable?
Let a,b be extended real numbers such that a{x∈R:a≤x≤b}⊆R is uncountable.
We can see that an uncountable set minus a countable set is indeed uncountable. Suppose for an uncountable set $A$ and a countable set $B$ that $A-B$ is countable. The union of countably many countable sets is countable; thus $(A-B) cup B$ is countable.
What happens if a set is uncountable?
A new correspondence can also be established between A and the union of all elements in A and B, since this is another countable set. Intuitively, it would seem even more obvious that the same would apply if A was uncountable. The larger set would still be uncountable.
Is (a ∖ B) ∪ B countable?
Assume that A ∖ B is countable. B is countable, so that would mean that ( A ∖ B) ∪ B is countable (finite union of countable sets is clearly countable). But then A ⊆ ( A ∖ B) ∪ B, so A is contained in a countable set, so it must itself be countable.
Is the Union of countably many countable sets countable?
The union of countably many countable sets is countable; thus ( A − B) ∪ B is countable. But then A is a subset of ( A − B) ∪ B and thus must be countable itself, which is a contradiction.