From every collection of objects, a person is able to make something called a subset. A subset is a smaller collection of objects made from a larger collection of objects. For example, Chris owns three Madonna albums. While Chris' collection is a set, we can also say that it is a subset of all Madonna albums.
Subsets
A Subset of set A is another set that contains only elements from the set A, but may not contain all the elements of A.
If B is a subset of A, we write it as $B \subseteq A$
A Proper Subset is a subset that is not identical to the original set, meaning it contains fewer elements than the original set.
If B is a proper subset of A, we write it as $B \subset A$
Example 1
Identify if the following sets are subsets, proper subsets, or neither of each other.
A = the set of all even numbers, $B=\{2,4,6\}$
A = the set of all even numbers, $C=\{2,3,4,6\}$
A = Primary Colors, B = Colors of the Rainbow
$A=\{\text{Red},\text{Blue},\text{Yellow}\}$, D = The set of Primary Colors
Solution
Looking at these two sets:
$$A=\{0,2,4,6,8,...\} \hspace{1cm} B=\{2,4,6\}$$
It can be seen that set B contains some of the elements of set A, but not all of the elements. This means that $B\subset A$, meaning B is a proper subset of A.
Looking at these two sets:
$$A=\{0,2,4,6,8,...\} \hspace{1cm} C=\{2,3,4,6\}$$
We can see that while $2, 4,$ and $6$ are all in set A, $3$ is not in set A which means that C is not a subset of set A. And looking at it backwards, since A contains elements that aren't in C, that means A is not a subset of set C either. We would denote this as $A\not\subset C$.
Set $A=\{\text{Red},\text{Blue},\text{Yellow}\}$, which is the set of primary colors, and set $B=\{\text{Red},\text{Orange},\text{Yellow},\text{Green},\text{Blue},\text{Indigo},\text{Violet}\}$, which is the set of all the colors of the rainbow. We can see that the colors Red, Blue, and Yellow are in both sets. In this case, Set B is the larger set, thus we would say that set A is a proper subset of set B. Written as $A\subset B$.
We just discussed that the set of primary colors is $D=\{\text{Red},\text{Blue},\text{Yellow}\}$, and $A=\{\text{Red},\text{Blue},\text{Yellow}\}$. Since these sets are equal sets, Set D is a subset of Set A, and Set A is a subset of Set D. This means that this is not a proper subset because they are equal sets. We would denote this symbolically with $A\subseteq D$. We could also write this the other way as $D\subseteq A$.
Try It
The Set $A=\{1,3,5\}$ is a subset of what larger set of numbers?
What if we wanted to create different subsets of a given set? How many subsets would we need, and how do we know how to find that?
Number of Subsets
The number of possible subsets that can be formed from a set of "$n$" elements can be found by $2^n$.
The number of possible proper subsets that can be formed from a set of "$n$" elements can be found by $2^n-1$.
Example 2
How many subsets and proper subsets can be found using set $A=\{\text{The colors of the rainbow}\}$? Identify an example of a subset of Set A that contains three elements.
Solution
First, we need to find the cardinality of Set A, where we'd find $n(A)=7$. Thus, the number of subsets would be $2^7=128$, and proper subsets would be $2^7-1=128-1=127$.
There are multiple subsets of Set A containing three elements that could be created, given that Set $A=\{\text{Red},\text{Yellow},\text{Green},\text{Blue},\text{Indigo},\text{Orange},\text{Violet}\}$. An example of a subset containing three elements could be something as simple as Set $B=\{\text{Red},\text{Yellow},\text{Orange}\}$. Once again, it is important to note that this is not the only possible answer for this question. There are approximately 35 different options just for this question.
Set Operations
Commonly, sets interact. For example, you and a new roommate decide to have a house party, and you both invite your circle of friends. At this party, two sets are being combined in a union of friends. As is with friendships, typically two people share other friendships, and we end up having this intersection of friends. These types of interactions amongst friends also happens in set theory. If you were to take all of the people of interest and put them in one giant group, that would be a universal set.
Universal Set
A universal set is a set that contains all the elements we are interested in. This should be defined by the context of the problem.
Example 3
Identify the universal set of the subsets below.
$A=\{2,4,6,8\}$
$B=\{\text{Red}, \text{White}, \text{Blue}\}$
$C=\{\text{Rock},\text{Rap},\text{Hip-Hop}\}$
Solution
The Universal set of set A would be the set of all even numbers.
The Universal set of set B would be the set of all colors.
The Universal set of set C would be the set of all music genres.
Set Operations
The union of two sets contains all the elements contained in either set (or both sets).
Denoted by $A\cup B$
More formally, $x\in A\cup B$ if $x\in A$ or $x\in B$
The intersection of two sets contains only the elements that both sets share in common.
Denoted by $A\cap B$
More formally, $x\in A\cap B$ if $x\in A$ and $x\in B$
The complement of a set is the set of all elements that are in the universal set but not in the set of interest. In short, it's what's not in A.
Denoted by $A^c$ or $A'$
Example 4
Consider the sets $A=\{\text{Red},\text{Green},\text{Blue}\}$, $B=\{\text{Red},\text{Yellow},\text{Orange}\}$, and $C=\{\text{Red},\text{Orange},\text{Yellow},\text{Green},\text{Blue},\text{Purple}\}$. Find:
$A\cup B$
$A\cap B$
$A'\cap C$
Solution
The symbol $\cup$ means union, which means we combine everything from A and everything from B into one giant set to get $A\cup B=\{\text{Red},\text{Green},\text{Blue},\text{Yellow},\text{Orange}\}$
The symbol $\cap$ means intersection, which tells us to only take the elements that set A and B have in common. Looking at both sets, if we highlight the elements they have in common, we'd see that only Red is in both sets.
$$A \cap B = \{\text{Red}\}$$
First, we need to find the complement of set A, which are all the colors that are not in A. If we think about the set of all colors, $A'$ is a very long set. Thus for this example, it may be easiest to think about which colors are in C that aren't in A and use that as our method. Since we are intersecting these two sets, we only want the ones they have in common. By doing it this way it would help us eliminate all those colors that could be in $A'$.
Thus $C=\{\text{Red},\text{Orange},\text{Yellow},\text{Green},\text{Blue},\text{Purple}\}$ and $A=\{\text{Red},\text{Green},\text{Blue}\}$. We can see that the colors orange, yellow, and purple are in set C, but not in Set A. This gives us our solution set for $A'\cap C=\{\text{Orange},\text{Yellow},\text{Purple}\}$