Subsets

We should have noticed that some sets encompass other sets. For example: $\mathbb{Z}$ encompass $\mathbb{N}$ ; $\mathbb{W}$ encompass $\mathbb{N}$; $\mathbb{Q}$ encompass $\mathbb{Z}$; and $\mathbb{R}$ encompass $\mathbb{Q}$.

In the same way, the set $\{1, 2, 3, 4, 5\}$ encompasses $\{1, 2, 3\}$. Every element of $\{1, 2, 3\}$ is also an element of $\{1, 2, 3, 4, 5\}$.

Another way to write this is that every element of $\mathbb{N}$ is also an element of $\mathbb{Z}$.

This relationship is called being a subset: all elements of one set are also elements of another set. We write this as $A \subseteq B$, which is read “A is a subset of B.”

We can write the subset relations mentioned above like this.

• $\mathbb{Q} \subseteq \mathbb{R}$

• $\mathbb{Z} \subseteq \mathbb{Q}$

• $\mathbb{W} \subseteq \mathbb{Z}$

• $\mathbb{N} \subseteq \mathbb{W}$

• $\{1,2,3\} \subseteq \mathbb{N}$

What if both sets encompass each other?

$\{1, 2, 3\}$ encompasses the set $\{3, 2, 1\}$ and vice versa. By definition, in such a case, we still use the term subset where both of the sets are equal.

Proper Subsets

We say that $A$ is a proper subset of $B$ if $A$ is a subset of $B$ but there is some element in $B$ that is not in $A$. This excludes the possibility that both sets encompass each other. A proper subset relation is presented as $A \subset B$ which is also read as "A is a proper subset of B".

We should note that the earlier examples we mentioned are all of them proper subsets.

Improper Subsets

Improper subsets are those sets which encompass each other which is just a fancy way of saying that both sets are equal. It is denoted by $A \subseteq B$ and is also read as "A is a subset of B". This also means that every set is also a subset of itself. $A \subseteq A$ is always true.

Modern Set Theory Practice

For the sake of convenience and standard, we generally use $A \subseteq B$ to cover both cases. $A \subseteq B$ means that $A$ could either be a proper subset or an improper subset of $B$. Similarly, you can write $\mathbb{N} \subseteq \mathbb{Z}$ or $\mathbb{N} \subset \mathbb{Z}$, both are correct.

The symbol $\subset$ says that this is strictly a proper subset while the symbol $\subseteq$ says that the sets can be either proper or improper.

Empty Set is a Subset of every set

Take some time to think why would we say that the empty set is a subset of every set. Generally, it is always correct to say $\phi \subseteq A$.

We need to look at the definition of subsets in the following way. We can say that $A$ is a subset of $B$ if there are no elements in $A$ that are also not in $B$. Clearly, the empty set does not have any elements that are also not in some other set. Thus, empty sets are a subset of all other sets. But why does it matter? Because it influences the size of the Power Set.

Venn Diagrams

Venn Diagrams are a great way to visualize sets. Venn diagrams are simple pictures used to represent sets and their relationships. They are drawn with shapes, usually circles, to show how groups overlap or differ. Each circle stands for a set, and the shaded parts show the results of set operations like union, intersection, and complement.

Explaining the Diagram:

The circles represent a given set. There are three circles that means that we are visualizing three sets here. When two circles overlap, it means that these sets share some common elements. If a circle is completely inside another circle, it means that all elements of the inner circle are also in the outer circle, thus the inner set is the subset of the outer set. In the given diagrams, we have the following relations.

• $C \subseteq B$ - Because all elements of C are also inside B.
• $B \not\subseteq C$ - Become some elements of B are not in C.
• $A \not\subseteq B$ - Become some elements of A are not in B.
• $B \not\subseteq A$ - Because some elements of B are not in A.

In this diagram, we see the hierarchy of the numbers sets and their relations. The outer rectangle represents the set of real numbers which contains all the rational and irrational numbers. And we see the following relations:

• $Q \subseteq R$ - Because all elements of Q are also inside R.
• $Z \subseteq Q$ - Because all elements of Z are also inside Q.
• $W \subseteq Z$ - Because all elements of W are also inside Z.
• $N \subseteq W$ - Because all elements of N are also inside W.

It also follows that the subset relations can be called out for deeply nested sets. Such as;

• $N \subseteq R$ - Because all elements of N are also inside R.

Hopefully this visualization helps you understand what subsets are. Venn diagrams will be further explained in the next lessons as new set operations are introduced.

Exercise Set 1: