Of course. If a subset $B$ of $A$ has cardinality strictly greater than the cardinality of $A$ itself, then there is an injection from $A$ to $B$, but not from $B$ to $A$, by the Cantor–Bernstein–Schroeder theorem. For an infinite setm, it is possible to have an injection into a proper subset, but there is also a trivial injection (inclusion) from a subset to the whole set.lamsung said:If a set has cardinality m then none of its subsets has cardinality greater than m.
Is it necessarily true for a infinite set case?
The cardinality of an infinite subset is the measure of the number of elements in that subset. In other words, it is a way to quantify the size of an infinite set.
The cardinality of an infinite subset is infinite, meaning that it cannot be counted or listed in a finite amount of time. In contrast, the cardinality of a finite set is a finite number that can be counted or listed.
Yes, it is possible for an infinite subset to have the same cardinality as its parent set. This occurs when the infinite subset is a proper subset of the parent set, meaning that it contains all the elements of the parent set but also has additional elements.
Yes, it is possible for two different infinite subsets to have the same cardinality. This is known as having a one-to-one correspondence, where each element in one subset can be paired with exactly one element in the other subset.
The cardinality of an infinite subset can be determined by finding a bijection, or a function that maps each element of the subset to a unique element of the parent set. This allows for a one-to-one correspondence between the elements of the subset and the elements of the parent set, indicating that they have the same cardinality.