Proper and Improper Subsets
In set theory, a subset can be either proper or improper. An improper subset is a subset that is equal to the original set, whereas a proper subset contains only some of the elements of the original set.
Every set A is always a subset of itself:
$$ A \subseteq A $$
Because A is equal to itself, it is considered an improper subset of A.
Note. Some introductory texts also classify the empty set as an improper subset. However, in modern set theory, the empty set is generally regarded as a proper subset of every non-empty set because it is distinct from the set itself.
When Two Sets Are Equal
A useful way to understand improper subsets is to look at two equal sets.
Suppose that A and B contain exactly the same elements.
In that case, every element of A belongs to B:
$$ A \subseteq B $$
At the same time, every element of B belongs to A:
$$ B \subseteq A $$
When both conditions are true, the two sets must be equal.
This means that each set is a subset of the other. Since neither set is strictly contained in the other, they are considered improper subsets of one another.
Note. A set A is called a proper subset of B if every element of A belongs to B and A is different from B. In other words, B must contain at least one element that is not in A.
Example
Consider the following sets:
$$ A = \{ 1,3,4 \} $$
$$ B = \{ 1,3,4 \} $$
Because the two sets contain exactly the same elements:
$$ A \subseteq B $$
$$ B \subseteq A $$
and therefore:
$$ A = B $$
Each set is an improper subset of the other.
The Empty Set
The empty set is the set that contains no elements.
It is represented by the symbol:
$$ \varnothing = \{ \} $$
One of the most important properties of the empty set is that it is a subset of every set.
At first glance, this may seem surprising. After all, how can a set with no elements be contained in every other set?
The answer lies in the definition of a subset. A set X is a subset of A if every element of X also belongs to A. Since the empty set has no elements, there is nothing that can violate this condition.
Example
Consider the empty set and the set:
$$ A = \{ 1,3,4 \} $$
Since the empty set contains no elements, every element of the empty set belongs to A. Therefore:
$$ \varnothing \subseteq A $$
A formal proof can be obtained by contradiction.
Assume that the empty set is not a subset of A.
If that were true, there would have to be at least one element of the empty set that does not belong to A.
However, the empty set has no elements.
This contradiction shows that the assumption is false. Therefore, the empty set must be a subset of A.
If A is non-empty, the empty set is also a proper subset of A because:
$$ \varnothing \subseteq A $$
and
$$ \varnothing \neq A $$
Alternative Proof. The union of a set A with any of its subsets B leaves the set unchanged: $$ A \cup B = A $$ The same happens when B is the empty set: $$ A \cup \varnothing = \{ 1,3,4 \} \cup \{ \} = \{ 1,3,4 \} = A $$ Since the union with the empty set does not add any new elements, this is consistent with the fact that the empty set is a subset of A.
