Cardinality
Cardinality is a set theoretic way to measure the ‘size’ of a set. If a set A has a larger cardinality than a set B, we intuit A to be a bigger set.
Cardinality is a set theoretic way to measure the ‘size’ of a set. If a set A has a larger cardinality than a set B, we intuit A to be a bigger set.
Any two non-empty open intervals I = (a,b) and J = (c,d) have the same cardinality.
Two sets have the same cardinality if there is a bijection between them. So we wish to find a bijective function f \colon I \to J.
We consider a simple linear function such that f(a) = c, f(b) = d. Geometrically we can consider stretching the interval (a,b) by a factor of d - c, and then shifting it to start at c. This gives the following function.
f(x) = c + (d-c)\frac{x - a}{b-a}.
We must now prove the function bijective. If f(x_1) = f(x_2), then by simple algebra we obtain x_1 = x_2. Let y \in J. Then there exist x \in I such that
a + \frac{y-c}{d-c}(b-a) = x.
Our function is bijective, and the open sets have the same cardinality.
Every interval has cardinality \mathfrak{c}.