Q.Let $A = \{1, 2, 3, 4, 5\}$, $B = \{x : x \text{ is a factor of } 4\}$, $C = \{1, 4, 9\}$. Insert the correct symbol '$\in$' or '$\notin$' in each of the following to make the statement true:

What Is a Set?

Imagine you have a handful of marbles. You can put them together in a little pile and say "these are my marbles." That pile — that collection — is a set. The marbles themselves are the elements (or members) of the set.

The whole idea is simple: a set is just a well-defined collection of distinct objects. "Well-defined" means there is no ambiguity about what belongs and what does not. If I say "the set of tall students in this class," that is not well-defined — how tall is "tall"? But if I say "the set of students whose height is at least 175 cm," that is precise. Every student either qualifies or does not.

The Intuition First

Think of a bag. You put things into it. The bag is the set; the things inside are the elements. The bag itself is not the same as its contents — it is the container, the idea of the collection. You can have an empty bag (the empty set, written $\emptyset$ or $\{\}$). You can have a bag with one thing (a singleton). You can have a bag with many things.

The order in which you put things into the bag does not matter. If you put in a red marble, then a blue one, or the blue one first — it is the same bag. Also, duplicates are irrelevant. If you put in a red marble twice, you still just have one red marble in the bag. Sets care only about what is in them, not how many times you list it.

The Precise Statement

A set is an unordered collection of distinct objects, called its elements or members. If $x$ is an element of the set $A$, we write $x \in A$. If $x$ is not an element, we write $x \notin A$.

Sets are usually denoted by capital letters ($A, B, C, \dots$) and their elements by lowercase letters ($a, b, c, \dots$). You can describe a set in two ways:

1. Roster (list) form: List the elements inside curly braces.

   Example: $A = \{1, 2, 3\}$ means the set containing the numbers 1, 2, and 3.

2. Set-builder form: State a property that all elements satisfy.

   Example: $B = \{x \mid x \text{ is a natural number and } x < 4\}$ means the same set $\{1, 2, 3\}$.

Why This Matters …