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\}#.
