Sets are collections of objects. They may be collections of mathematical objects, such as numbers or functions. They may be collections of any other type of object such as students in a class or times of day. We can even have sets of sets.
It is useful to understand some general notation when working with sets. We will work with sets more explicitly in later sections. For now, we really just need to be able to understand the notation.
We will usually use capital letters for sets, such as \(S\) or \(A\text{.}\) If we want to talk about elements in a set \(S\text{,}\) we use the notation \(x\in S\). We read this notatation as “\(x\) is in \(S\)” or “\(x\) is an element of \(S\text{.}\)” If \(x\) is not in \(S\text{,}\) then we use the notation \(x\notin S\).
If we want to list the specific elements of a set, we use curly brackets, \(\{ \}\text{,}\) around the elements of the set. We can also do this with a description of the elements in the set.
We use the notation \(B\subseteq A\) for “\(B\) is a subset of \(A\)”. If \(B\) is not a subset of \(A\text{,}\) then we use the notation \(B\nsubseteq A\). Note, some books use \(A\subset B\) as the notation for subset.
It is important to understand the difference between subset, \(\subseteq\text{,}\) and element, \(\in\text{.}\) For example, if \(S=\{1, 2, 3, 4, 5\}\text{,}\) then \(1\in S\text{,}\) but \(1\nsubseteq S\text{.}\) This is because 1 is an element, not a set. Similarly, \(\{1\}\subseteq S\text{,}\) but \(\{1\}\notin S\text{.}\) This is because \(\{1\}\) is a set, not an element. In general, when working with sets, you should identify the elements of the set. Then sets of those elements are subsets. The curly brackets are our way of saying “set.”
\(\mathbb{R}\), the set of real numbers. These are all the numbers your are familiar with from calculus: whole numbers, positives, negatives, fractions, decimals, square roots, \(e\text{,}\)\(\pi\text{,}\) etc.
\(\mathbb{Q}\), the set rational numbers. These are all the whole numbers and fractions: positive, negative, and zero. We will revisit this set in more detail later.
\(\mathbb{N}\), the set of natural numbers. These are all the positive whole numbers. Some books include zero, some do not. Since this can be confusing, we will avoid this notation in this class (but you might see it in future classes). Instead, we will use one of the next two notations, which more clearly denote inclusion of zero, or not.
It is important to be able to distinguish between elements of a set and subsets of a set. Determine if each of the following is true or false. If it is false, what small change in notation would make it true?
