Can a subset be bigger than the original set?

The set of all subsets of a given set is bigger than the set itself. Let there be a set X. We say that X is bigger than Y if they are not equivalent but there exists a 1-1 correspondence between Y and a subset of X. If X is bigger than Y then Y is smaller than X.

Does a subset of a set have the same cardinality?

cardinality. This shows that a proper subset of a set can have the same cardinality as the set itself. (b) The function f : N → {2, 3, 4,… } defined by f(n) = n + 1 for n ∈ N is a bijection, so the set of natural numbers N = {1, 2, 3,… } has the same cardinality as its proper subset {2, 3, 4,… }.

Do you think there are sets whose cardinality is actually larger than that of the set of real numbers?

> Is there any set with a cardinality greater than the Real numbers? Yes. Cantor’s Theorem [ https://en.wikipedia.org/wiki/Cantor’s_theorem ] shows that the power set (the set of all subsets) of any set has a strictly greater cardinality than the set itself.

Is there a set larger than the real numbers?

The set of real numbers (numbers that live on the number line) is the first example of a set that is larger than the set of natural numbers—it is ‘uncountably infinite’. There is more than one ‘infinity’—in fact, there are infinitely-many infinities, each one larger than before!

Can a subset be larger?

So a subset cannot have a larger cardinality than the set. No. Consider U, a non-empty subset of V. (For the empty subset U of V, U has cardinality 0, so since cardinality is always greater than or equal to 0, #U is less than or equal to #V.)

What do we call the set with the same cardinality?

The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets.

Is there a set having the largest cardinality?

Cantor’s theorem implies that there are infinitely many infinite cardinal numbers, and that there is no largest cardinal number.