Tao Analysis I - 3.6.2

Exercise 3.6.2

Show that a set $X$ has cardinality $0$ if and only if $X$ is the empty set.

Let's remind ourselves of the definitions of cardinality.

Def 3.6.1 (Equal cardinality) We say that two sets $X$ and $Y$ have equal cardinality iff there exists a bijection $f : X \to Y$ from $X$ to $Y$.

Def 3.6.5 Let $n$ be a natural number. A set $X$ is said to have cardinality $n$, iff it has equal cardinality with $\{i ∈ N : 1 ≤ i ≤ n\}$. We also say that $X$ has $n$ elements iff it has cardinality $n$.

We need to show both of the following:

• If a set is the empty set then it has cardinality 0.
• If a set has cardinality $0$, it must be the empty set.

Show $X = \emptyset \implies \#X = 0$

We assume $X=\emptyset$. The only set it has a bijection with is the empty set (link). This existence of a bijection is sufficient for Def 3.6.1. By Def 3.6.5 this cardinality is $0$, because the empty set has a bijection with the set $\{i ∈ N : 1 ≤ i ≤ n\}$ with $n=0$.

Thus we have shown $X = \emptyset \implies \#X = 0$.

Show $\#X = 0 \implies X = \emptyset$

If a set $X$ has cardinality $0$, then by Def 3.6.5 it has cardinality equal to to the set $\{i ∈ N : 1 ≤ i ≤ n\}$ where $n=0$. This is the empty set.

Thus we have shown $\#X = 0 \implies X = \emptyset$.

By showing both statements, we have shown that a set $X$ has cardinality $0$ if and only if $X$ is the empty set. $\square$