Unit 1: Further on Sets

Unit 1 explores advanced concepts of set theory, building upon the foundational operations and definitions introduced in lower grades. It establishes rigorous methods for describing sets, identifying set relationships, and performing complex operations such as symmetric difference and Cartesian products. Students will learn to apply these principles using Venn diagrams and numerical formulas to solve practical mathematical problems.

1.1 Sets and Elements

• A set is a collection of well-defined objects or elements.
• A set is well-defined if it is possible to determine whether a given object belongs to the set or not.
• Elements (or members) are the objects within a set, usually represented by small letters (a, b, c).
• Sets are typically denoted by capital letters (A, B, C, X, Y, Z).
• The Greek symbol $\epsilon$ (epsilon) denotes “belongs to” ($a \in A$).
• The symbol $\notin$ denotes “does not belong to” ($b \notin A$).

1.2 Set Description

• Verbal method (Statement form): The well-defined description of elements is written as an ordinary English language statement.
• Listing Methods:
  • Complete listing method (Roster Method): All elements are listed, separated by commas, and enclosed in set braces { }.
  • Partial listing method: Used when listing all elements is difficult; a few elements are listed followed by three dots (…) to indicate the pattern.
• Set builder method (Method of defining property): Describing a set by writing the condition or property that its members must satisfy.
• Standard Number Sets:
  • Natural numbers ($\mathbb{N}$): {1, 2, 3, …}.
  • Whole numbers ($\mathbb{W}$): {0, 1, 2, 3, …}.
  • Integers ($\mathbb{Z}$): {…, -3, -2, -1, 0, 1, 2, 3, …}.

Process: Describing a Set Using the Set Builder Method

1. Open the set brace { .
2. Write a variable (e.g., $x$) to represent the elements.
3. Draw a vertical line ( | ) or colon ( : ) which means “such that.”
4. State the specific condition or property the variable must satisfy.
5. Close the set brace } .

1.3 The Notion of Sets

• Empty Set (void/null set): A set which does not contain any element, denoted by { } or Ø.
• Finite Set: A set where the number of elements is limited and countable.
• Infinite Set: A set where the elements continue without end.
• Equal Sets: Two sets A and B are equal ($A=B$) if and only if they have exactly the same elements.
• Equivalent Sets: Two sets are equivalent ($A \leftrightarrow B$) if there is a one-to-one correspondence between them, meaning $n(A) = n(B)$.
• Universal Set (U): A set containing elements of all related sets in a discussion without repetition.
• Subset ($\subseteq$): Set A is a subset of set B if every element of A is also an element of B.
  • Any set is a subset of itself ($A \subseteq A$).
  • The empty set is a subset of every set.
  • The number of subsets for a set with $n$ elements is $2^n$.
• Proper Subset ($\subset$): Set A is a proper subset of B if $A \subseteq B$ and $A \neq B$.
  • The number of proper subsets for a set with $n$ elements is $2^n – 1$.
• Superset ($\supset$): If $A \subseteq B$, then B is a superset of A.

1.4 Operations on Sets

• Set operation: The process of creating new sets from existing ones.
• Venn diagram: A pictorial representation showing sets as interlocking circles within a rectangle (the universal set).
• Union ($\cup$): The set of all elements that are either in A, in B, or in both.
• Intersection ($\cap$): The set of all elements that are in both A and B.
• Disjoint Sets: Two sets are disjoint if their intersection is empty ($A \cap B = \emptyset$).
• Absolute Complement ($A’$): The set of all elements in the universal set U that are not in set A.
• Difference of Sets ($A-B$ or $A \backslash B$): The set of all elements in A that are not in B; also called the relative complement.
• Symmetric Difference ($A \Delta B$): The set defined as $(A \backslash B) \cup (B \backslash A)$ or $(A \cup B) \backslash (A \cap B)$.
• Cartesian Product ($A \times B$): The set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

1.5 Application

• Number of Elements Formula: For finite sets A and B, the number of elements in their union is: $n(A \cup B) = n(A) + n(B) – n(A \cap B)$.
• Special Case: if $A \cap B = \emptyset$, then $n(A \cup B) = n(A) + n(B)$.

Process: Finding the Number of Elements in an Intersection

1. Identify the values for $n(A)$, $n(B)$, and $n(A \cup B)$.
2. Use the formula $n(A \cup B) = n(A) + n(B) – n(A \cap B)$.
3. Rearrange to solve for the unknown: $n(A \cap B) = (n(A) + n(B)) – n(A \cup B)$.
4. Subtract the total union from the sum of individual sets to find the overlap.

Key Terminology

• Absolute complement: The set of all elements of U that are not in A.
• Cartesian product: The set of all ordered pairs (a, b) where $a \in A$ and $b \in B$.
• Complete listing method: A method where all elements of a set are listed within braces and separated by commas.
• Difference of sets: The set of elements in A that are not in B.
• Disjoint: Two sets whose intersection is the empty set.
• Elements: The objects belonging to a set.
• Empty set: A set which does not contain any element.
• Equal sets: Sets that have exactly the same elements.
• Equivalent sets: Finite sets that have the same number of elements.
• Intersection: The set of elements common to both sets.
• Ordered pairs: Pairs where the order of components is significant, used in Cartesian products.
• Partial listing method: Listing some elements of a set followed by three dots to show the pattern.
• Proper subset: A subset that is not equal to the original set.
• Relative complement: Another name for the difference between two sets.
• Set: A collection of well-defined objects or elements.
• Set-builder method: Describing a set by a property its members must satisfy.
• Subset: A set where every element is also an element of another set.
• Symmetric difference: Elements in either A or B, but not both.
• Union: Elements that are in set A, set B, or both.
• Universal set: A set containing all elements of related sets.
• Venn diagram: A pictorial representation of sets using interlocking circles.