Subsets and Complements - Definition & Examples

(Video) What is a Subset?

by Don't Memorise

This video by Don't Memorise covers sets and subsets with some real life examples.

$\large{\text{Summary}}$

Suppose that you took a trip to a museum (let's call it museum $A$) that specialized in Egyptian pottery, and then took a trip to the Metropolitan Museum of Art.

In this situation, you would note that all of the kinds of things that are present in museum $A$ are also present in the Met; we would say that museum $A$ covers a subset of the genres of art that museum $B$ does.

It works the same way with numbers.

Say that we have some set that contains the numbers $1, 2,$ and $3$; let's call it set $A$. Now, suppose that we also have some other set called set $B$ that contains the numbers $1, 2, 3, 4,$ and $5$. Since all of the elements that are contained in set $A$ are also contained in set $B$, we can say that set $A$ is a subset of set $B$, and we can write that in mathematical notation like so: $$A\ \subseteq B$$

Also, returning to my first example, notice how museum $A$ and the Met do not both contain the same exact genres of art. Likewise, in my second example, our two sets do not contain the same exact elements. This means that our set $A$ is a strict subset of set $B$, and we can express that like so: $$A\ \subset B$$

You can think of the "subset of" and "strict subset of" symbols as being analogous to the symbols for "less than or equal to" and "less than": a bar underneath the symbol dictates that the two items being compared are allowed to be "equal," and the absence of this bar dictates just the opposite.

Finally, let's talk about how the empty set, notated mathematically as $\varnothing$, fits into all of this.

Suppose you have a bowl of fruit containing three apples, two bananas and one orange. We can express this fruit bowl as a set like so: $$\left\{a,a,a,b,b,o\right\}$$

Now, suppose we remove each piece of fruit from the bowl until we are left with an empty bowl. Since this leftover emptiness is in fact part of the collection of fruits that were initially inside of the bowl, we can say that "nothing," or the empty set, is a subset of the set we described above: $$\varnothing\subseteq\left\{a,a,a,b,b,o\right\}$$

Going by this logic, we can say that "nothing," or the empty set, is in fact a subset of all sets, including both of the sets that we discussed prior to this example.

Let's start with the $1$ number subsets: $$\begin{align}\{1\}\\ \{2\}\\ \{3\}\\ \{4\}\\ \end{align}$$ Now the $2$ number subsets: $$\begin{align}\{1,2\}\\ \{1,3\}\\ \{1,4\}\\ \{2,3\}\\ \{2,4\}\\ \{3,4\}\\ \end{align}$$ $3$ number subsets: $$\begin{align}\{1,2,3\}\\ \{1,2,4\}\\ \{1,3,4\}\\ \{2,3,4\}\\ \end{align}$$ The last two subsets are the complete set, and the empty set: $$\begin{align}\{1,2&,3,4\}\\ \{&\varnothing\} \end{align}$$

In total there are $16$ subsets for this set.

(Video) Subsets and Proper Subsets

by HCCMathHelp

In this video, HCCMathHelp explains how to find subsets and proper subsets.

$\large{\text{Summary}}$

Let's talk terminology.

Subsets: A subset is a set that's entirely made of elements that are also in another set. If you wanted use math notation to say that $A$ was a subset of $B$, you would write $A \subseteq B$.

Supersets: A superset is a set that contains all of the elements that are in a smaller set. If $B$ has all of the elements of $A$ in it, and then maybe some other ones that aren't in $A$, we can say that $B$ is a superset of $A$. (We can also say that $A$ is a subset of $B$.)

Proper Subsets: A proper subset is a subset that doesn't just equal its superset. In other words, it's a subset with fewer elements than its superset. The symbol for a proper subset is $\subset$.

That might have gotten weird. Let's try it out with a couple of sets: $$\begin{align} A &= \{2, 3, 4, 5, 6\}\\ B &= \{2, 3, 4, 5, 6\}\\ C &= \{2, 4, 6\} \end{align}$$

Subsets and Complements

First, we'll discuss subsets; then, complements.

Subsets

Before we talk about subsets, you should get yourself familiar with sets themselves.

A subset is part of a set. All of the elements in a subset must be in the original set.

A null set is a set with no elements.

Consider the set of shapes pictured above. Here are some sample subsets (in picture form). The subsets are in the blue, transparent circles.

Notice how the subsets are only made up of things that were in the original set. There's nothing extra added in. The number of subsets is $3$, and the number of elements in the set is $9$. This is a finite set. The elements of the set are all shapes.

Now, let's try with sets written in roster form:

Proper subsets are like examples in the problem above—they contain some but not all elements in the original set.

Improper subsets are still subsets, but they contain every element that the original set does. They are equal to the original set.

There are symbols used to represent each of these types of subsets:

It's like a "C", but longer.

Notice the line underneath it! It looks just like the "or equal to" signs ($\le, \ge$), but for subsets.

Both chocolate and vanilla are in the original set, but they don't make up the whole original set. This means that it is a proper subset.

Complements

The complement of a set is all of the values not included in the set.

If you combine the elements in a set and its complement, you end up with something called the universal set.

The most common ways to represent a complement of a set (let's call this set A) is to write $A^c$ or $A'$.

In the picture above, the purple circle represents set A, the gray box around the circle represents the complement of set A, and the two put together (i.e, the whole picture) represent the universal set (written as U).

If you take all the elements of $A$ out of $U$, you're left with $\{2, 6, 7\}$, which means that's the complement of $A$ in $U$.

If $A$ and $A^c$ are complements, $U$ must be all the elements of both of them combined (in other words, the union of $A$ and $A^c$). Combining $A$ and $A^c$ gets us: $$U = \{\text{red, orange, yellow, green, blue}\}$$

One thing to understand is that a set can be a subset of itself. For example, $\mathbb{R}$ is a subset of $\mathbb{R}$.

To take this concept one step further, we examine intersection of sets and visual venn diagrams. An application of sets is in statistics, but you'll learn about that later!

Let $A, B$ be sets. We say that $A$ is a subset of $B$ (which we can write as $A \subseteq B$) if all the elements in $A$ are also elements of $B$.

We can write this all as $A \subseteq B$ if $\forall x \in A$, $x \in B$.

• The $\subseteq$ symbol means "is a subset".
• The upside-down A ($\forall$) means "for all".
• The weird "E" looking symbol ($\in$) means "exists in".

Example:

Let's define a set that we'll call $A$. $$A = \{1, 2, 3, \{4, 5\}\}$$

• The set $\{1, 2\}$ is a subset of $A$.
• The set $\{1, 2, 3\}$ is another subset of $A$
• The set $\{1, 2, 3, \{4, 5\}\}$ is ALSO a subset of $A$

But wait! The set $\{4, 5\}$ is NOT a subset of $A$, because $\{4\}$ and $\{5\}$ aren't in $A$ by themselves. Instead, they're in another set that's PART of set $A$.

So we can say $\{4, 5\} \in A$ (the set $\{4, 5\}$ is in $A$) or {{$4, 5$}} $\subseteq A$ (the set made up of the set $\{4, 5\}$ is a subset of $A$), but we CAN'T say $\{4, 5\} \subseteq A$.

On the other hand, we can say both that $2 \in A$ (the number $2$ is part of $A$) and that the set $\{2\}$ is a subset of $A$.

Test Yourself!

Alright, let's try it out on two sets! $$\begin{align} B &= \{1, 3, \{1, 2\}\}\\ C &= \{1, 2\} \end{align}$$

Note: The symbol $\subset$ is occasionally used to denote a proper subset, i.e. a subset which is not equal to the set. For example, $\{1,2\} \subset \{1,2,3\}$. Additionally, $\{1,2,3\} \subseteq \{1,2,3\}$, but $\{1,2,3\} \not\subset \{1,2,3\}$ because the two sets are equal.