Can an infinite set have an infinite subset?

An infinite set can definitely have infinitely many subsets, as there are infinite objects in that set, which are alone a subset by themselves. The interesting thing is that the number of subsets in this infinite set are *infinitely greater* than the number of objects in the set itself.

How many subsets does an infinite set have?

In general, a set with N elements has 2N subsets. This works when you get to infinite sets and their cardinal numbers too. 23=8 subsets.

Can an infinite set be countable?

An infinite set is called countable if you can count it. For example, the even numbers are a countable infinity because you can link the number 2 to the number 1, the number 4 to 2, the number 6 to 3 and so on.

How do you prove something is countably infinite?

A set is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers. In other words, one can count off all elements in the set in such a way that, even though the counting will take forever, you will get to any particular element in a finite amount of time.

Is there an infinite subset of a finite set?

Hence, for any finite set F, there does not exist an infinite subset I. There is actually a proof you can probably find which does the same thing, just it takes a different angle: Prove that every subset of a finite set is finite.

How do you prove a set is infinite?

Definition: A set is infinite, if it can’t be mapped one-to-one with an n-element set for any natural number n. Lemma (can be proven using the principle of induction): If a set is infinite, then it has an n-element subset for every natural number n. Let X be any infinite set.

How do you prove that an empty set is finite?

The empty set ∅ = { } ⊆ X for every set (finite or infinite) X, and | ∅ | = 0 is finite. Use the definition of a subset to show that for all sets X, ∅ ⊆ X. (Hint: look up the definition of “vacuously true”, if you are stuck.)

How do you find the two-element subset of a set?

So there is a two-element subset U 2 = { x 1, x 2 } of distinct elements of S. For each N ∈ N, S contains an element distinct from each element in U N = { x 1, x 2, ⋯, x n }, so define U N + 1 = U N ∪ { x N + 1 } where x N + 1 is an element of S distinct from each element of U N.