# Equivalent Sets

# Why

We want to talk about the size of a set.

# Definition

Two sets are equivalent
if there exists a bijection between them.
Let $X$ be a set.
Then set equivalence as a relation in
$\powerset{X}$ is an equivalence relation (see Equivalence Relations).

## Notation

If $A$ and $B$ are sets and they are
equivalent, then we write $A \sim B$, read
aloud as “$A$ is equivalent to $B$.”

## Basic result

Every set is equivalent to itself, whether two
sets are equivalent does not depend on the
order in which we consider them, and if two
sets are equivalent to the same set then they
are equivalent to each other.
These facts can be summarized by the following
proposition.

Let $X$ a set. Then $\sim$ is an equivalence
relation on $\powerset{X}$.

# For natural numbers

Every proper subset of a natural number is
equivalent to some smaller natural number.

# Equivalence to subsets

It is unusual that a set can be equivalent to
a proper subset of itself.

A set may be equivalent to a proper subset of
itself.
The example is the set of natural numbers and
the function $f(n) = n^+$.
It is a bijection from $\omega $ onto $\N $.

However, this never holds for natural numbers.
If $n \in \omega $ then $n \not\sim x$ for
any $x \subset n$ and $x \neq n$.