Subset

What Is a Subset?

A subset is a set where all of its elements are also contained within another, larger set. Think of it like a collection of items taken from a bigger collection. If every item you chose is from the original group, then your new group is a subset of the original.

Let's use an analogy. Imagine a set called Fruit which contains $\{\text{apple, orange, banana, grape}\}$. If you create a new set called MySnack with the elements $\{\text{apple, grape}\}$, then MySnack is a subset of Fruit. Why? Because every element in MySnack (apple, grape) is also found in the Fruit set.

In mathematics, we use a special symbol to show this relationship: $\subseteq$. So, we would write:

$\text{MySnack}\subseteq \text{Fruit}$

This is read as "MySnack is a subset of Fruit."

For this relationship to be true, there can't be any element in the smaller set that isn't in the larger set. If we had a set $\text{MyLunch}=\{\text{apple, sandwich}\}$, it would not be a subset of Fruit because the element 'sandwich' is not in the Fruit set. We show this using the symbol $⊈$:

$\text{MyLunch}⊈\text{Fruit}$

This fundamental idea of one set being contained within another is a building block for many other topics in mathematics.

How Do You Identify a Subset?

To determine if one set is a subset of another, you just need to perform a simple check. For a set $A$ to be a subset of a set $B$, you must go through every single element in $A$ and confirm that it also exists in $B$.

Let's follow a clear process:

1. Identify the two sets. Let's call them $A$ (the potential subset) and $B$ (the larger set).
2. Take the first element of set $A$. Check if this element is present in set $B$.
3. If it is, move to the next element in $A$.
4. If it is not, you can stop immediately. $A$ is not a subset of $B$.
5. Repeat the process until you have checked every element in $A$. If you get through the entire list without finding any elements that aren't in $B$, then you can confidently say that $A\subseteq B$.

What Is the Difference Between Proper and Improper Subsets?

The term "subset" has two more specific variations: proper subsets and improper subsets. Understanding the difference is key to being precise in your mathematical language.

Proper Subset ( $\subset$ )

A set $A$ is a proper subset of set $B$ if $A$ is a subset of $B$, but $A$ is not equal to $B$. This means that set $B$ must contain at least one element that is not in set $A$. The symbol for a proper subset is $\subset$, which looks like the subset symbol without the "or equal to" line underneath. Think of it like the difference between $<$ (less than) and $\le$ (less than or equal to).

• Let $B=\{1,2,3\}$.
• The set $A=\{1,2\}$ is a proper subset of $B$ because all elements of $A$ are in $B$, and $B$ has an element ($3$) that is not in $A$. We write $A\subset B$.

Improper Subset

An improper subset is a subset that contains every single element of the original set. This means the only improper subset of a set $B$ is the set $B$ itself. It's a subset because all its elements are in the original set (of course!), but it's not "proper" because it's not smaller.

• Let $B=\{1,2,3\}$.
• The set $C=\{1,2,3\}$ is an improper subset of $B$. We can still write $C\subseteq B$ because the "or equal to" part of the symbol is satisfied. However, we cannot write $C\subset B$.

Here is a table to summarize the differences:

What Is the Special Role of the Empty Set?

In set theory, there is a very special set called the empty set (or null set). It is the set that contains no elements at all. It is represented by the symbol $\mathrm{\varnothing }$ or by empty curly braces $\{\}$.

The empty set has a unique and fundamental property: The empty set is a subset of every set.

This might seem strange at first. How can a set with nothing in it be a subset of, say, $\{a,b,c\}$? Let's think back to the definition. For a set $A$ to be a subset of $B$, every element in $A$ must also be in $B$. The only way for this to be false is if you can find an element in $A$ that is not in $B$.

Let's try to prove that $\mathrm{\varnothing }$ is not a subset of $\{a,b,c\}$. To do this, we would need to find an element inside $\mathrm{\varnothing }$ that is not inside $\{a,b,c\}$. But the empty set has no elements! It's impossible to find such an element. Since we cannot prove the statement false, it must be true. This is sometimes called a "vacuously true" statement in logic.

So, for any set $S$ you can imagine, it is always true that:

$$\mathrm{\varnothing }\subseteq S$$

This is a rule you must always remember when working with subsets. When you are asked to list all the subsets of a set, your list must always begin with the empty set, $\mathrm{\varnothing }$.

How Many Subsets Does a Set Have?

Manually listing every subset of a large set would be tedious and prone to errors. Thankfully, there is a simple and powerful formula to calculate the total number of subsets a set has without listing them.

The number of subsets depends on the number of elements in the set. The number of elements in a set $A$ is called its cardinality, denoted as $|A|$ or $n(A)$. Let's use $n$ to represent the number of elements.

In this formula, $n$ is the number of elements in the original set.

Why Does This Formula Work?

Let's consider a set $S=\{a,b,c\}$. Here, $n=3$. To build any subset, we have to make a decision for each element: do we include it or not?

• For element 'a': We have 2 choices (Yes, include it / No, don't include it).
• For element 'b': We also have 2 choices (Yes / No).
• For element 'c': We again have 2 choices (Yes / No).

The total number of possible combinations of these choices is found by multiplying the number of options for each element: $2\times 2\times 2=2^3=8$. Each unique combination of choices creates a unique subset. For example, choosing "Yes" for 'a', "No" for 'b', and "Yes" for 'c' gives us the subset $\{a,c\}$. Choosing "No" for all three gives us the empty set $\mathrm{\varnothing }$.

If you want to find the number of proper subsets, you just subtract $1$ (for the set itself). So the number of proper subsets is $2^n-1$.

How Do You Find All Subsets of a Given Set?

Knowing the number of subsets is great, but sometimes you need to list all of them. To do this without missing any, it's best to use a systematic approach. The best method is to list the subsets based on their size (number of elements).

Here is a step-by-step method:

1. Start with size 0: There is only one subset with zero elements: the empty set ($\mathrm{\varnothing }$).
2. List subsets of size 1: Create a subset for each individual element in the original set.
3. List subsets of size 2: Combine every possible pair of elements from the original set.
4. Continue this process, increasing the size by one each time, until you reach...
5. List subsets of size $n$: There is only one subset with $n$ elements: the original set itself.

This organized method ensures you don't accidentally skip a combination, which is easy to do if you list them randomly. For a set with $4$ elements, you would follow the same pattern: list the 1 subset of size 0, the 4 subsets of size 1, the 6 subsets of size 2, the 4 subsets of size 3, and the 1 subset of size 4, for a total of $1+4+6+4+1=16$ subsets.

Common Mistakes to Avoid

The concept of subsets is straightforward, but a few common mistakes can trip students up. Being aware of these pitfalls is the best way to avoid them.

1. Confusing Elements ($\in$) with Subsets ($\subseteq$)
   The symbol $\in$ means "is an element of," while $\subseteq$ means "is a subset of." An element is a single item, while a subset is a set. Let $S=\{5,6,7\}$.
   Correct: $5\in S$
   Correct: $\{5\}\subseteq S$
   Incorrect: $5\subseteq S$ (A number can't be a subset, only a set can.)
   Incorrect: $\{5\}\in S$ (The set containing 5 is not an element of S.)
2. Forgetting the Empty Set
   When asked to list all subsets, the most commonly forgotten one is the empty set, $\mathrm{\varnothing }$. Remember, it is a subset of every set, so it should always be the first one on your list.
3. Forgetting the Set Itself
   The second most forgotten subset is the original set itself. Every set is an improper subset of itself. This should always be the last subset on your list when organizing by size.
4. Using the Wrong Subset Symbol
   Be careful with the distinction between $\subseteq$ (subset) and $\subset$ (proper subset). If two sets might be equal, you must use $\subseteq$. Only use $\subset$ when you are certain the subset is smaller than the larger set. For example, $\{1,2\}\subset \{1,2,3\}$ is correct, but $\{1,2,3\}\subset \{1,2,3\}$ is false.
5. Miscalculating the Number of Subsets
   A common arithmetic error is calculating $2^n$ as $2\times n$. For a set with $4$ elements, the number of subsets is $2^4=16$, not $2\times 4=8$. Always remember it's an exponent.

Quick Summary

Here are the most important points to remember about subsets:

• Definition: A set $A$ is a subset of a set $B$ if every element of $A$ is also an element of $B$.
• Subset Notation: $A\subseteq B$ means $A$ is a subset of $B$.
• Proper Subset Notation: $A\subset B$ means $A$ is a subset of $B$ but is not equal to $B$.
• The Empty Set: The empty set, $\mathrm{\varnothing }$, is a subset of every set.
• The Improper Subset: Every set is a subset of itself.
• Formula for Number of Subsets: A set with $n$ elements has $2^n$ subsets.
• Formula for Proper Subsets: A set with $n$ elements has $2^n-1$ proper subsets.