Sets

A set is just a collection of distinct elements—think of it as a mathematical container with clearly defined boundaries. Whether it holds numbers, objects, or ideas, sets are the foundation of modern mathematics! 🧺

Sets

A set is a collection of things.

The things are called elements of a set.

Colors = \{\text{Red}, \text{Blue}, \text{Green}\}

Numbers = \{1,2,3\}

Sets can be finite or infinite.

Some\ Even\ Integers = \{2,4,6\}

All\ Even\ Integers = \{..., -4, -2, 0, 2, 4, ...\}

Concept Check 1

Is the set of prime numbers infinite or finite?

The set of prime numbers is infinite.

The set of prime numbers is infinite by Euclid’s theorem. Assume finitely many primes p_1, \dots, p_n; then P = p_1 \cdots p_n + 1 is not divisible by any p_i, a contradiction. For example:

2 \times 3 + 1 = 7 is prime, 2 \times 3 \times 5 + 1 = 31 is prime, and so on.

The number of elements in a set is called the cardinality.

|Colors| = 3

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

A = \{2,4,6\}, B = \{4,2,6\}, C = \{4,2,7\}

A = B A \neq C

To express that 2 is an element of A, we denote:

2 \in A \text{2 exists in A}

5 \notin A \text{5 does not exist in A}

Some sets are so significant that we reserve special symbols for them:

\emptyset = \{\} \quad \textbf{(empty set)}

\mathbb{N} = \{1, 2, 3, ... \} \quad \textbf{(natural numbers)}

\mathbb{Z} = \{..., -2, -1, 0, 1, 2, ... \} \quad \textbf{(integers)}

\mathbb{R} = \{..., -0.22,...,0,...,1,..., \pi, ... \} \quad \textbf{(real numbers)}

We visualize \mathbb{R} as an infinitely long number line.

Set-Builder Notation

A special notation called set-builder notation is used to describe sets that are too big or complex to list between braces.

All\ Even\ Integers_{1} = \{..., -4, -2, 0, 2, 4, ...\}

All\ Even\ Integers_{2} = \{2x: x \in \mathbb{Z} \}

\text{The set of all numbers of the form } 2x \text{ such that } x \in \mathbb{Z}

All\ Even\ Integers_{1} = All\ Even\ Integers_{2}

Concept Check 2

Which of the following set-builder notation represents this set?

\{ 2, 4, 8, 16, 32, 64, ... \}

The correct representation is \{ 2^x: x \in \mathbb{N} \}, as it includes all natural numbers x where 2^x generates the elements of the set.