Definition2.1.1Set
A set is an unordered collection of objects.
Section2.1Basic concepts
¶During the 19th century, mathematicians became increasingly concerned with the foundations of mathematics and formalizing logic. Late in that century, Georg Cantor started considering sets and their properties. We have the following informal definition of a set.
Because a set is unordered, it does not make any sense to say that an element of a set occurs twice. We generally denote finite sets with curly braces, such as
\begin{equation*} \{ 1,2,3,4,5 \}. \end{equation*}For large or infinite sets, it helps to use set builder notation. For example, the even integers can be expressed as
\begin{equation*} \{ x \mid x = 2k, k \in \mathbb{Z} \}. \end{equation*}This should be read as “the set of all \(x\) such that \(x = 2k\) for some integer \(k\text{.}\)” It is the vertical bar | that is interpreted as “such that,” but it may also be read as “with the property,” or “satisfying,” or “for which,” and probably even others. Note that some authors (including those of your textbook, prefer to use a colon : instead of a vertical bar; you should get comfortable with both notations. We can simplify with shorthand the form of the set of even integers we wrote above as
\begin{equation*} \{ 2k \mid k \in \mathbb{Z} \}. \end{equation*}Also, whenever the elements of your collection lie in some larger set, you should specify this in the set builder notation. For example, our description of the even integers really should have specified that \(x\) itself is an integer, as in
\begin{equation*} \{ x \in \mathbb{Z} \mid x = 2k, k \in \mathbb{Z} \}. \end{equation*}Set can contain just about anything, including other sets! For example, the set \(A =\{0,\{1,2\},3\}\) has as its elements the integers, \(0,3\) as well as the set \(\{1,2\}\text{.}\) The set \(A\) has three elements.
Definition2.1.2Empty Set
The empty set is the set with no elements. In particular, the empty set is the unique set for which the statement \((\forall x, x \notin \emptyset)\) is true.
Definition2.1.3Subset
A set \(A\) is said to be a subset of \(B\text{,}\) denoted \(A \subseteq B\text{,}\) if every element of \(A\) is an element of \(B\text{.}\) Logically, this means
\begin{equation*} A \subseteq B \iff (\forall x)(x \in A \implies x \in B). \end{equation*}We also say that \(A\) is contained in \(B\text{,}\) or \(B\) contains \(A\text{.}\)
The logical form of \(A \subseteq B\) tells us how to prove it. This statement is a universally quantified implication (conditional), so to prove it, we pick any element \(x\) of the set \(A\) and prove that it is also an element of \(B\text{.}\)
Theorem2.1.4
For any sets, \(A,B,C\text{,}\)
- \(\emptyset \subseteq A\text{.}\)
- \(A \subseteq A\text{.}\)
- If \(A \subseteq B\) and \(B \subseteq C\text{,}\) then \(A \subseteq C\text{.}\)
Definition2.1.5Set equality
Two sets \(A,B\) are equal if they have exactly the same elements. Logically, this means
\begin{equation*} A = B \iff (\forall x)(x \in A \iff x \in B). \end{equation*}Since a biconditional is equivalent to the conditional in both directions, this means
\begin{equation*} A = B \iff (A \subseteq B \wedge B \subseteq A). \end{equation*}The above definition tells us we may prove set equality two ways. Either, we may prove both subset relations, or wee may string together a bunch of if and only if statements.
Definition2.1.6Power set
Given a set \(A\text{,}\) we form form the power set of \(A\text{,}\) denoted \(\mathscr{P}(A)\text{,}\) which is the set containing all subsets of \(A\text{.}\) In set builder notation this is:
\begin{equation*} \mathscr{P}(A) = \{ X \mid X \subseteq A \}. \end{equation*}Suppose \(A = \{1,2,3\}\text{,}\) then the power set is:
\begin{equation*} \mathscr{P}(A) = \big\{\emptyset, \{1\}, \{2\}. \{3\}. \{1,2\}, \{2,3\}, \{1,3\}, \{1,2,3\} \big\}, \end{equation*}which has 8 elements. We will prove later that if a finite set has \(n\) elements, then its power set has \(2^n\) elements.
Remark2.1.8
By Theorem 2.1.4, we can see that for any set \(A\text{,}\) the power set \(\mathscr{P}(A)\) always contains \(\emptyset\) and \(A\text{.}\) Note also that \(X \in \mathscr{P}(A)\) if and only if \(X \subseteq A\text{.}\) You will have to be careful not to confuse subsets with elements; the notions are different.