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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.

Describing sets

One can describe a set by


  1. The set of all even integers is given by \(\{ 2n : n \text{ is an integer }\}\).

  2. The set of all polynomials in \(x\) with real coefficients having degree at most two is given by \(\{ ax^2 + bx + c : a,b,c \in \mathbb{R}\}\).

Special sets

Sets of \(n\)-tuples

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, \(\mathbb{Z}^3 = \left \{ \begin{bmatrix} a\\ b\\ c\end{bmatrix} : a, b, c \in \mathbb{Z}\right\}\).

Common Set Notation

Let \(A\) and \(B\) be sets.

Quick Quiz


  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 \(A \backslash B\).  

  2. Which of the following are members of \(\mathbb{Q} \cap \{ a : a \in \mathbb{R}\) and \(a \gt \sqrt{2}\}\)?

    1. 0  

    2. 5  

    3. \(\sqrt{3}\)