11.1 Set
A set is a collection of objects with well-defined common characteristics.
Representing Sets
Sets can be described using three main methods:
- Description: Writing a statement in words. Example: $A$ is the set of prime numbers less than 10.
- Roster / Listing Method: Listing all elements within curly brackets separated by commas. Example: $A = \{2, 3, 5, 7\}$.
- Set Builder Notation: Using a variable and property rule. Example: $A = \{x : x \text{ is a prime number and } x < 10\}$.
Elements and Number of Elements
- An object that belongs to a set is called an element of the set, represented by the symbol $\in$.
- If an object does not belong to a set, it is written using the symbol $\notin$.
- The total number of elements in set $A$ is denoted by $n(A)$.
Empty Set
- A set that contains no elements is called an empty set (or null set).
- Represented by the symbol $\phi$ or $\{\}$. (Note: $\{\phi\}$ or $\{0\}$ are NOT empty sets).
- The number of elements in an empty set is zero, $n(\phi) = 0$.
11.2 Venn Diagrams, Universal Set, Complement of a Set, and Subset
Universal Set ($\xi$)
A universal set is a set that contains all the elements under consideration for a particular discussion, represented by the symbol $\xi$.
Complement of a Set ($A'$)
The complement of set $A$, written as $A'$, is the set containing all elements in the universal set $\xi$ that do NOT belong to set $A$.
Venn Diagrams
A Venn diagram is a visual representation where:
- The universal set $\xi$ is represented by a rectangle.
- Sets inside $\xi$ are represented by circles or other closed shapes.
- Elements are written inside their corresponding regions with dots preceding each element.
Subset ($\subset$)
- Set $A$ is a subset of set $B$, written as $A \subset B$, if every element of set $A$ is also an element of set $B$.
- If $A$ is not a subset of $B$, it is written as $A \not\subset B$.
- Important Properties:
- The empty set $\phi$ is a subset of every set ($\phi \subset A$).
- Every set is a subset of itself ($A \subset A$).
- For a set with $n$ elements, the total number of possible subsets is given by $2^n$.