Sets, Subsets and Elements

3. Probability: Randomness

Sets, Subsets and Elements

Before introducing the concept of randomness, it is important to define the terms set, element, and subset.
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Sets and Elements

Definition

A set is a collection of distinct objects, considered as an object in its own right. An object can be anything from a number, a letter, a set or a combination of these.

The distinct objects in a set are called the elements of that set.

Notation

$\text{set}=\{\text{element}, \text{element}, \ldots\}$

For example, the numbers $2$ , $4$ and $6$ are distinct objects when considered separately. But when they are considered collectively, they form a single set with three elements, written as $\{2,4,6\}$ .

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Sets of elements are often depicted with the use of a Venn Diagram. Sets are usually given a name (e.g. $A$ ) and are depicted as a circle. The elements of the set (e.g. $1, 2, 3, 4, 5, 6$ ) are then put within the circle. The Venn Diagram below corresponds to the set $A = \{1, 2, 3, 4, 5, 6\}$ .

The set $B$ is a subset of $A$ if $B$ is contained inside of $A$ . That is, all elements of $B$ are also elements of $A$ .

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Subsets are displayed in a Venn diagram as a circle within a circle. The image below shows the set $A = \{1, 2, 3, 4, 5, 6\}$ and its subset $B = \{1, 3, 5, 6\}$ .