Chapter 1 Section 1

Sets

  • A Set is collection of things, called the “elements” of the set.
  • Two sets are the same means they have exactly the same elements. Knowing the elements means knowing the set.

Describing sets by listing elements

A set can be described by listing its elements using curly braces.

\[ A = \{1,2,3\} \] means \(A\) is the set whose elements are \(1\), \(2\), and \(3\). The symbols \(\{\) and \(\}\) are special and are used to describe sets.

Note: The sets \(A=\{1,2,3\}\) and \(B=\{3,1,2\}\) are the same because they have the same elements. So we write \(A=B\).

The \(\in\) symbol

The symbol \(\in\), which looks a little like a backwards \(3\) and a little like a greek \(\epsilon\), means “is an element of.”

  • \(1\in A\) means \(1\) is an element of the set \(A\).

The symbol \(\not\in\) means “is not an element of.”

  • \(5\not\in A\) means that \(5\) is not an element of \(A\).

Basic examples

  • The natural numbers \(\mathbb{N}\) is the set of counting numbers \(1,2,3,\ldots\) \[ \mathbb{N} = \{1,2,3,\ldots\} \]

  • The integers are the are the positive and negative whole numbers, and zero: \[ \mathbb{Z} = \{\ldots, -5,-4,-3,-2,-1,0,1,2,3,\ldots\} \]

  • The rational numbers \(\mathbb{Q}\) are the positive and negative fractions and zero.

We take for granted addition, multiplication, commutative, associative, laws etc.

Other sets

  • the alphabet
  • the set of words in English
  • the set of people now living
  • the set of chairs in my house (what’s a chair….)

The empty set

There is exactly one set which nas no elements, called the empty set.
The empty set can be written \(\emptyset\) or \(\{\}\).

The cardinality of a set

  • If \(A\) is a set, we write \(|A|\) for the number of elements in the set if that number is finite.

  • If \(A=\{1,2,3\}\) then \(|A|=3\)

  • We will study cardinality in more detail at the end of the class; for now, we will take this idea for granted. We also take for granted that a set like \(\mathbb{Z}\) has infinitely many elements.

The real numbers

  • The real numbers is the set of all numbers with possibly infinite decimal expansions (positive or negative). A proper definition is hard to give and is usually done in analysis. We will work with the real numbers informally as we did in Calculus.