Set Theory

A set is a collection of objects, called elements.
We list its elements between curly brackets.

List all possible results when rolling a standard die .

• An element is an object contained in a set.
• \(\in\) means "is an element of" or "belongs to".
• \(\notin\) means "is not an element of" or "does not belong to".

\(2\in \{1,2,3,4,5,6\}\) and \(7\notin \{1,2,3,4,5,6\}\).

Two sets are equal if they have exactly the same elements.

Determine if the sets \(\{2,6,4\}\) and \(\{2,4,6\}\) are equal.

Determine if the sets \(\{1,2,3\}\) and \(\{1,2,4\}\) are equal.

The empty set is a set with no elements. It is written as \(\{\}\) or \(\emptyset\).

The set of natural numbers, denoted \(\N\), is the set of counting numbers starting from zero:$$\N = \{0, 1, 2, 3, 4, 5, 6, \dots\}$$

A set \(A\) is a subset of a set \(B\) if every element in \(A\) is also in \(B\). We write this as \(A \subseteq B\).

Is \(A \subseteq B\) when \(A = \{2, 4, 6\}\) and \(B = \{1, 2, 3, 4, 5, 6\}\)?

The set-builder notation is a way to describe a set by giving a rule that its elements must follow. It is written like this:$$\{x \in E \mid \text{condition on } x\}$$In this notation, \(x\) represents an element from a set \(E\), and the symbol \(\mid\) (or sometimes \(:\)) separates \(x\) from the condition it must meet. It reads: “the set of all \(x\) in \(E\) such that \(x\) follows the given condition.”

Let \(E\) be the set of results from rolling a standard six-sided die .
What are the elements of the set \(\{x \in E \mid x \text{ is even}\}\)?

An ordered pair, denoted \((a, b)\) or sometimes \(ab\), is a pair of elements where the order matters.
More generally, an ordered n-tuple, denoted \((a_1, a_2, \dots, a_n)\), is a sequence of \(n\) elements where the order of the elements is significant. An ordered pair is a special case when \(n = 2\).

In a sprint relay race, two runners are paired up. Let \(L\) be Louis and \(H\) be Hugo. The ordered pair \((L, H)\) means Louis runs first, then passes the baton to Hugo. The ordered pair \((H, L)\) means Hugo runs first, then passes to Louis. These are different races.

\(\Card{A}\) denotes the number of elements in the set \(A\).

\(\Card{\{1,2,3,4,5,6\}}=6=\)

• A finite set has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting.
• An infinite set is not a finite set.

• \(\{1, 2, 3\}\) is finite because it has exactly 3 elements .
• \(\N=\{0,1, 2, 3, \dots\}\) is infinite because the counting goes on forever without stopping.

A universal set is the set of all elements considered.

The complement of a set \(A\), denoted \(A'\), consists of all elements in \(U\) that are not in \(A\). Sets \(A\) and \(A'\) are said to be complementary.

Given the universe \(U = \{1, 2, 3, 4, 5, 6\}\) and the set \(A = \{1, 3, 5\}\), find the complement \(A'\).

The intersection of two sets \(A\) and \(B\), written \(A \cap B\), is the set of elements that are in both \(A\) and \(B\).

What is the intersection \(\{1, 2, 3\} \cap \{2, 3, 4\}\)?

The union of two sets \(A\) and \(B\), written \(A \cup B\), is the set of all elements in \(A\) or \(B\) (or both).

What is the union \(\{1, 2, 3\} \cup \{2, 3, 4\}\)?

A Venn diagram uses a rectangle to show the universal set \(U\) and circles to represent other sets within it.

Here’s a Venn diagram for \(U = \{1, 2, 3, 4, 5, 6\}\) and \(A = \{1, 3, 5\}\):

This table shows common set operations and their Venn diagrams:

Notation	Meaning	Venn Diagram
\(A\)	Set \(A\)
\(A'\)	Complement of \(A\) (everything in \(U\) not in \(A\))
\(A \subseteq B\)	\(A\) is a subset of \(B\)
\(A \cup B\)	Union of \(A\) and \(B\) (all elements in \(A\) or \(B\))
\(A \cap B\)	Intersection of \(A\) and \(B\) (elements in both)
\(A \cap B = \{\}\)	\(A\) and \(B\) are disjoint (no common elements)

Venn diagrams help solve problems by showing the number of elements in each region.

In a Venn diagram, we use brackets around numbers to show how many elements are in each region.

Consider this Venn diagram:Here, there are 6 elements in both \(A\) and \(B\), 4 in \(A\) but not \(B\), 5 in \(B\) but not \(A\), and 1 outside both. Total elements: \(A\) has \(4 + 6 = 10\), \(B\) has \(6 + 5 = 11\).

Number sets are groups of numbers defined by specific properties, and they form the foundation of mathematics. We start with the simplest set, the natural numbers (\(\N\)), which we use for basic counting. From there, we build the integers (\(\Z\)) by adding negative numbers to include opposites. Next, we expand to the rational numbers (\(\Q\)) by allowing fractions, which cover division between integers. Finally, we reach the real numbers (\(\R\)) by filling in all the gaps with irrational numbers, creating a complete number line. Below are the most common number sets you’ll encounter, each one growing from the previous set in this progression.

• The natural numbers, denoted \(\N\), are the counting numbers starting from zero:$$\N = \{0, 1, 2, 3, 4, 5, 6, \dots\}$$
• The integers, denoted \(\Z\), include all whole numbers—positive, negative, and zero:$$\Z = \{\dots, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, \dots\}$$
• The rational numbers, denoted \(\Q\), are numbers that can be written as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). Examples include \(\frac{1}{2}\), \(-3\), and \(\frac{5}{4}\):$$\Q = \left\{ \frac{p}{q} \mid p, q \in \Z, q \neq 0 \right\}$$
• The real numbers, denoted \(\R\), include all numbers on a continuous number line, such as rational numbers (like \(\frac{1}{3}\)) and irrational numbers (like \(\sqrt{2}\) and \(\pi\)):$$\R = \{\text{all numbers, rational and irrational}\}$$

The number sets are nested as follows:$$\N \subset \Z \subset \Q \subset \R$$This means every natural number is an integer, every integer is a rational number, and every rational number is a real number.

An interval is a set of all real numbers between two endpoints, which may or may not be included in the set.

The set of all real numbers between 0 and 1, including 1 but not 0, is an interval. It is written as \(\{x \in \R \mid 0 < x \leq 1\}\).

Intervals are often shown on a number line using these conventions:

1. An \emph{open circle} (or parenthesis) means the endpoint is not included.
2. A \emph{filled circle} (or bracket) means the endpoint is included.
3. An \emph{arrow} shows that the interval extends to positive or negative infinity.

The number line representation of \(\{x \in \R \mid 0 < x \leq 1\}\) is:

Interval Notation	Set-builder notation	Number line representation
\(\CloseBracketLeft a, b \CloseBracketRight\)	\(\{x \in \R \mid a \leqslant x \leqslant b\}\)
\(\CloseBracketLeft a, b \OpenBracketRight\)	\(\{x \in \R \mid a \leqslant x < b\}\)
\(\OpenBracketLeft a, b \CloseBracketRight\)	\(\{x \in \R \mid a < x \leqslant b\}\)
\(\OpenBracketLeft a, b \OpenBracketRight\)	\(\{x \in \R \mid a < x < b\}\)
\(\CloseBracketLeft a, +\infty\OpenBracketRight \)	\(\{x \in \R \mid a \leqslant x\}\)
\(\OpenBracketLeft a, +\infty \OpenBracketRight \)	\(\{x \in \R \mid a < x\}\)
\(\OpenBracketLeft -\infty, a\CloseBracketRight\)	\(\{x \in \R \mid x \leqslant a\}\)
\(\OpenBracketLeft -\infty, a\OpenBracketRight\)	\(\{x \in \R \mid x < a\}\)