Set Theory explores the fundamental concepts of sets, including operations, relations, and applications in mathematics.
Some sets, like the set of all natural numbers \(\mathbb{N} = {1, 2, 3, \ldots}\), go on forever. These are infinite sets.
The cardinality of a set is a measure of its size, or how many elements it has. For finite sets, it's just the count. But infinite sets get interesting!
Not all infinite sets are the same size. The set of natural numbers is infinite, but so is the set of real numbers between 0 and 1. Surprisingly, there are more real numbers than natural numbers!
Mathematician Georg Cantor showed that some infinities are bigger than others using clever arguments like diagonalization.
The set of all even numbers is infinite, just like the set of all whole numbers.
The set of real numbers is a 'bigger' infinity than the set of natural numbers.
Infinite sets can have different sizes, and cardinality helps us compare them.