It is natural for us to classify items into groups, or sets, and consider how those sets overlap with each other. We can use these sets to understand different relationships between groups and to analyze survey results. As with everything in mathematics, we have terms and words that have special meanings. This unit is derived around a much more complex mathematics called set theory. In set theory, the objects that are sorted into groups are called elements, and a group of elements are known as a set.
Identify the set of all states in the United States that start with a "V".
Thinking through all the states in the United States, the ones that start with "V" are Vermont and Virginia.
However, what if I was to ask you to list all mothers? Is that something you can easily do? Not really, because this set is not well-defined. The set is not well-defined because it isn't specific enough. A mother could be a lot of things. It could be a step-mother, a biological mother, a human, an animal, and so on. The different types of descriptors goes on and on. Simply saying the set of all mothers doesn't help us break down the group of what could be a mother.
Identify if each of the following are well-defined sets or not. Why?
When it comes to listing sets, there are multiple ways in which a person can list out a set. The primary way we are going to list sets is using something called the roster method. With the roster method, each element of the set is put into a list-like format with a comma between each element and a set of curly brackets surrounding the list. An example of roster method for example 2.1 would be $\{0,2,4,6,8,10,12,...\}$. In this case, we use the $...$ to signify that the set goes on forever.
When writing and developing sets, the order is not overly important; it is just simply specifying the content. The number of distinct elements that fall into a set is known as the set cardinality. If we were talking about the cardinality of set A, it would be denoted as $n(A)$.
Identify the cardinality of each set given below.
When talking about set cardinality, there is always the possibility that a set does not have a definite cardinality. In that case, we call the set infinite. An example of an infinite set would be something like the set of all odd numbers. If we were to write that out, we could start to identify it as $O=\{1,3,5,7,9,...\}$. Note how the end of this set ends with the "..." signifying that the set goes on indefinitely. When a set is classified as finite, it has a definite cardinality.
Part of the reason we break elements into sets is so we can make comparisons between the sets. Two sets are said to be equivalent if they have the same cardinality, denoted by $A\cong B$. Two sets are classified as equal if they contain the same exact elements, denoted by $A=B$.
Identify if the sets below are equal, equivalent, or neither.
What would happen if we were to talk about the set of all unicorns? While some of us would love to believe unicorns exist, but sadly they do not. This means we'd have an empty set.
The empty set, also referred to as the null set, is the set that contains no elements. The empty set/null set is denoted by either $\{\}$ or $\varnothing$. The cardinality of this set is zero.
One last remark about notations before the close of this section would be about the symbol $\in$. This symbol's literal meaning is "element of" and it can be used in place of words. For example, should you want to use it to explain an element of set $A=\{\text{Red},\text{Yellow},\text{Blue}\}$, we could say "$\text{Blue}\in A$", and this would mean Blue is an element of set A.