Think of a set as a collection of distinct objects, called its elements. The objects might be numbers, points, or people. Membership tells us whether a particular object belongs to the set. In this lesson, we will work with small sets of numbers.

Imagine the set of students enrolled in a class. To check membership, ask whether a particular student is enrolled. Being first or last on the attendance list makes no difference to membership.

A set holds some things and not others, and that is the whole of what it does. With $A=\{2,4,6\}$: $2\in A$, and $5\notin A$ however close to the ring it is drawn.

We treat set and membership as basic notions in this course; the collection picture is an intuitive description. For a small finite set, we can list all its elements between braces, separated by commas. In the example below, A names the set whose elements are 2, 4, and 6.

• $4 \in A$ means "4 is an element of A." The symbol $\in$ expresses membership.
• $5 \notin A$ means "5 is not an element of A." The symbol $\notin$ expresses non-membership.

$M = \{3,8\}$. Then $3 \in M$.

The number 3 appears in M, so the statement says that 3 is an element of M.

Both sides contain 1, 3, and 5, with no other elements. The symbol = says that the sets are equal.

The distinct numbers are 4 and 9. This set has two elements, even though the first notation lists three numbers.

$\varnothing = \{\}$, but $\varnothing \ne \{0\}$.

The symbol ∅ denotes the empty set. The symbol ≠ means "is not equal to." Zero is an element of the set on the right.

A set can be given by listing its elements or by naming a condition they satisfy. The second form is the only one available once the set is infinite.

• A set is decided by its elements alone: order and repetition are invisible.
• $\in$ relates an object to a set; a set may be listed or described by a property.
• $\varnothing$ has no elements, and $\{\varnothing\}$ has one.
• $\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}$ name the number systems this course works in.

Great work! You now know how to:

• Read and write membership statements
• Describe a set by listing it or by naming a property
• Tell the empty set from a set whose only element is empty
• Name the four number systems $\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}$