Definition 2.1.1.1.
A set is an unordered collection of objects, called elements or members of the set.
We write \(b \in A\) which we read “\(b\) is an element of \(A\)” to denote that \(b\) is an element of the set \(A\text{.}\)
Subsection 2.1.1 Basic terminology
Definition 2.1.1.2.
The universal set is the set that contains all objects that are relevant for a specific discussion.
In our discussions of sets and set operations the domain of all propositional functions and thus for all quantification will be the universal set, unless specified explicitly.
Definition 2.1.1.4.
Two sets \(A\) and \(B\) are equal , denoted \(A = B\text{,}\) if they have the same elements, that is, \(\forall x (x \in A \iff x \in B)\text{.}\)
By the convention stated above \(\forall x (x \in A \iff x \in B)\) is read
“For all elements \(x\) of the universal set we have that \(x\) is an element of the set \(A\) if and only if \(x\) is an element of the set \(B\text{.}\)”
There are several ways to describe a set.
We specify a set in roster form by listing the elements explicitly between braces, for example, \(\set{1, 2, 7}\text{.}\) To indicate that the list of continues in a pattern that can be deduced from the preceding list of elements we use ellipsis (\(\ldots\)), for example
\begin{equation*}
\set{1, 2, 3,\dots, 10}=\set{1,2,3,4,5,6,7,8,9,10}\text{.}
\end{equation*}
When we specify a set in set builder notation we characterize the elements of the set by stating properties they must have to be members, for example,
\begin{equation*}
\set{x \given x \text{ is an odd positive integer less
than }10}=\set{1,3,5,7,9}.
\end{equation*}
Example 2.1.1.5. Set or element.
We formulate the sets of numbers that we often use in set builder notation.
Recall that set of integers is \(\ZZ=\set{\dots, -2, -1, 0, 1, 2, \dots}.\)
The set of natural numbers or positive integers is \(\NN=\set{x\given x\in\ZZ\land x\gt 0}=\set{1, 2, 3,\dots}.\)
The set of non-negative integers is \(\NN_0=\set{x\given x\in\ZZ\land x\ge 0}=\set{0, 1, 2,\dots}\text{.}\)
The set of rational numbers is \(\QQ=\set{\frac{p}{q} \given p\in \ZZ \land q\in\ZZ \land q \neq 0}.\)
We denote the set of real numbers by \(\RR\text{.}\) , For our purposes, the intuitive idea that they are all numbers that can be written in finite or infinite decimal representation is sufficient.
Example 2.1.1.6. Intervals.
We have special notation for intervals of real numbers.
\begin{align*}
[a,b] \amp = \set{x \given x \in \RR, a\leq x\leq b } \\
(a,b] \amp = \set{x \given x \in \RR, a \lt x\leq b } \\
[a,b) \amp = \set{x \given x \in \RR, a\leq x \lt b } \\
(a,b) \amp = \set{x \given x \in \RR, a \lt x \lt b }
\end{align*}
Example 2.1.1.7. Equality.
To show two sets are equal, we use subsets as described later in Proof Technique Proof Technique 2.1.2.7. It is more straightforward to show two sets are not equal. Since set equality is defined as the universal quantification (“for all”) of a biconditional (“if and only if”), to show two sets are not equal, it is enough to find any counterexample to the biconditional. In particular, we just need to produce any element that they do not share.
Proof Technique 2.1.1.8. To show \(A \neq B\).
Suppose \(A\) and \(B\) are sets, and we want to prove \(A\) is not equal to \(B\text{.}\)
-
Find a particular element \(a\) in \(A\) that is not a member of \(B\text{;}\) or find a particular element \(b\) in \(B\) that is not a member of \(A\text{.}\)
-
Conclude \(A \neq B\text{.}\)
Definition 2.1.1.9.
The empty set, denoted \(\emptyset\text{,}\) is the set with no elements.
Example 2.1.1.10. Two formulations for the empty set.
The empty set is \(\set{}\text{.}\)
That is, \(\emptyset=\set{}\)
