Sets are arguably the most fundamental objects in modern mathematics. Familiarity with set notation is a prerequisite to reading post-secondary mathematics. What follows is a brief summary of key definitions and concepts related to sets required in this course.
A set is a well-defined collection of distinct mathematical objects. The objects are called members or elements of the set.
specifying a rule or a verbal description. For example, one can say “let \(A\) be the set of all odd integers”. Then \(A\) is a set and its elements are all the odd integers.
enclosing the list of members within curly brackets. For example, \(C = \{2,4,5\}\) denotes a set of three numbers: 2, 4, and 5, and \(D = \{(2,4), (-1,5)\}\) denotes a set of two pairs of numbers.
Abbreviations can be used if the set is large or infinite. For example, one may write \(\{1,3,5,\ldots,99\}\) to specify the set of odd integers from \(1\) up to \(99\), and \(\{4,8,12,\ldots\}\) to specify the (infinite) set of all positive integer multiples of \(4\).
Another option is to use set-builder notation: \(F = \{n^3 : n\) is an integer with \(1\leq n \leq 100\}\) is the set of cubes of the first \(100\) positive integers. (The way to interpret this is as follows: \(F\) is the set of all \(n^3\) such that \(n\) is an integer from \(1\) to \(100\).)
The set of all even integers is given by \(\{ 2n : n \text{ is an integer }\}\).
The set of all polynomials in \(x\) with real coefficients having degree at most two is given by \(\{ ax^2 + bx + c : a,b,c \in \mathbb{R}\}\).
\(\mathbb{N}\) denotes the set of natural numbers; i.e. \(\{1,2,3,\ldots\}\).
\(\mathbb{Z}\) denotes the set of integers; i.e. \(\{\ldots,-2,-1,0,1,2,\ldots\}\).
\(\mathbb{Q}\) denotes the set of rational numbers (the set of all possible fractions, including the integers).
\(\mathbb{R}\) denotes the set of real numbers.
\(\mathbb{C}\) denotes the set of complex numbers. (This set will be introduced more formally later.)
There is a convenient notation for specifying sets of \(n\)-tuples whose entries are from the same set.
Let \(A\) be a set. Let \(n\) be a positive integer. Then, the set of \(n\)-tuples whose entries are elements of \(A\) is denoted by \(A^n\).
For example, \(\mathbb{Z}^3\) is the set of all \(3\)-tuples whose entries are integers. In other words, \(\mathbb{Z}^3 = \left \{ \begin{bmatrix} a\\ b\\ c\end{bmatrix} : a, b, c \in \mathbb{Z}\right\}\).
Let \(A\) and \(B\) be sets.
\(\lvert A \rvert\), called cardinality of \(A\), denotes the number of elements of \(A\). For example, if \(A = \{ (1,2), (3,4) \}\), then \(\lvert A \rvert = 2\).
\(A = B\) if and only if they have precisely the same elements. For example, if \(A = \{4, 9\}\) and \(B = \{ n^2 : n = 2 \) or \( n = 3 \}\), then \(A = B\).
\(A \subseteq B\) if and only if every element of \(A\) is also an element of \(B\). We call \(A\) a subset of \(B\). For example, \(\{1,8, 1107\} \subseteq \mathbb{N}\).
\(a \in A\) means \(a\) is a member of \(A\). For example, \(5 \in \mathbb{Q}\)
\(a \notin A\) means \(a\) is not a member of \(A\). For example, \(\frac{2}{7} \notin \mathbb{Z}\)
\(A \cap B\) denotes the set containing elements that are in both \(A\) and \(B\). \(A \cap B\) is called the intersection of \(A\) and \(B\). For example, if \(A = \{1,2\}\) and \(B = \{2,3\}\), then \(A \cap B = \{2\}\).
\(A \cup B\) denotes the set containing elements that are in either \(A\) or \(B\) or both. \(A \cup B\) is called the union of \(A\) and \(B\). For example, if \(A = \{1,2\}\) and \(B = \{2,3\}\), then \(A \cup B = \{1,2,3\}\).
\(A \backslash B\) denotes the set containing elements that are in \(A\) but not in \(B\). \(A \backslash B\) is read as “\(A\) drop \(B\)”. For example, if \(A = \{1,2\}\) and \(B = \{2,3\}\), then \(A \backslash B = \{1\}\).
Let \(A = \{1,3,5,7\}\) and \(B = \{0,3,6,7,9\}\). Write out the following sets: \(A\cup B\), \(A \cap B\), and \(A \backslash B\).
Which of the following are members of \(\mathbb{Q} \cap \{ a : a \in \mathbb{R}\) and \(a \gt \sqrt{2}\}\)?
0
5
\(\sqrt{3}\)