Preview Activity 1.1.
Suppose we try to define a set to be a collection of elements. So, by definition, the elements are the objects contained in the set. We use the symbol \(\in\) to denote that an object is an element of a set. So \(\notin\) means an object is not in the set — if \(x\) is an object in a set \(S\) we write \(x \in S\text{,}\) and if \(x\) is not an object in a set \(S\) we write \(x \notin S\text{.}\) We write sets using set brackets. For example, the set \(\{a,b,c\}\) is the set whose elements are the symbols \(a\text{,}\) \(b\text{,}\) and \(c\text{.}\) We can also include in the set notation conditions on elements of the set. For example, \(\{x \in \R : x \gt 0\}\) is the set of positive real numbers. We typically use capital letters to denote sets. Some familiar examples of sets are \(\R\text{,}\) the set of real numbers; \(\Q\text{,}\) the set of rational numbers; and \(\Z\text{,}\) the set of integers. Sets can also contain sets as elements. For example, the power set of a set \(S\) is the set of subsets of \(S\text{.}\) So the power set of \(S = \{1,2\}\) is the set \(\{\emptyset, \{1\}, \{2\}, \{1,2\}\}\text{.}\) (We will define subsets and the empty set later in this activity).
(a)
Consider the following “set” \(S\text{,}\) which is a collection of objects:
\begin{equation*}
S = \{A \text{ is a set } \mid A \notin A\}\text{.}
\end{equation*}
That is, \(S\) is the collection of sets that do not have themselves as elements. Given any object \(x\text{,}\) either \(x \in S\) or \(x \notin S\text{.}\)
(i)
Is \(S\) an element of \(S\text{?}\) Explain.
(ii)
Is it the case that \(S \notin S\text{?}\) Explain.
(iii)
Based on your responses to parts (a) and (b), explain why our current concept of a set as a collection of elements is not a good one.
(b)
Assume that we have a working definition of a set. In this part of the activity we define a subset of a set. The notation we will use is \(A \subset S\) if \(A\) is a subset of \(S\) that is not equal to \(S\text{,}\) and \(A \subseteq S\) if \(A\) is a subset of \(S\) that could be the entire set \(S\text{.}\) We will also say that \(A\) is contained in \(S\) if \(A\) is a subset of \(S\text{,}\) and call the relation \(A \subset S\) (or \(A \subseteq S\)) a containment.
(i)
How should we define a subset of a set? Give a specific example of a set and two examples of subsets of that set.
(ii)
If \(A\) is a set, is \(A\) a subset of \(A\text{?}\) Explain.
(iii)
What is the empty set \(\emptyset\text{?}\) If \(A\) is a set, is \(\emptyset\) a subset of \(A\text{?}\) explain.
