🧠Thinking Like a Mathematician Unit 4 – Set Theory and Relations in Math

Key Concepts and Definitions

• Sets are collections of distinct objects or elements
• Elements can be numbers, symbols, or other objects
• Set theory provides a foundation for modern mathematics
• Relations define connections or associations between elements of sets
• Functions are special types of relations where each input has a unique output
• Cartesian product of two sets $A$ and $B$ is the set of all ordered pairs $(a,b)$ where $a \in A$ and $b \in B$
• Cardinality of a set refers to the number of elements in the set
  • Finite sets have a specific number of elements
  • Infinite sets have an unlimited number of elements

Set Notation and Operations

• Sets are denoted using curly braces $\{\}$ with elements separated by commas
  • Example: $A = \{1, 2, 3, 4\}$
• Empty set or null set is represented by $\emptyset$ or $\{\}$
• Union of sets $A$ and $B$ is written as $A \cup B$ and includes all elements from both sets
• Intersection of sets $A$ and $B$ is written as $A \cap B$ and includes elements common to both sets
• Difference of sets $A$ and $B$ is written as $A - B$ or $A \setminus B$ and includes elements in $A$ but not in $B$
• Complement of set $A$ is written as $A'$ or $A^c$ and includes all elements in the universal set that are not in $A$
• Symmetric difference of sets $A$ and $B$ is written as $A \Delta B$ and includes elements in either $A$ or $B$ but not both

Types of Sets

• Universal set, denoted by $U$, contains all elements under consideration in a given context
• Subset $A$ is contained within another set $B$, written as $A \subseteq B$
  • Every set is a subset of itself, $A \subseteq A$
• Proper subset $A$ is contained within set $B$ but is not equal to $B$, written as $A \subset B$
• Superset $B$ contains all elements of another set $A$, written as $B \supseteq A$
• Power set of a set $A$, denoted by $\mathcal{P}(A)$, is the set of all subsets of $A$
  • Example: If $A = \{1, 2\}$, then $\mathcal{P}(A) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\}$
• Disjoint sets have no elements in common, i.e., their intersection is the empty set

Set Properties and Theorems

• Commutative property: $A \cup B = B \cup A$ and $A \cap B = B \cap A$
• Associative property: $(A \cup B) \cup C = A \cup (B \cup C)$ and $(A \cap B) \cap C = A \cap (B \cap C)$
• Distributive property: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ and $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
• Identity laws: $A \cup \emptyset = A$ and $A \cap U = A$
• Complement laws: $A \cup A' = U$ and $A \cap A' = \emptyset$
• Idempotent laws: $A \cup A = A$ and $A \cap A = A$
• De Morgan's laws: $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$

Relations and Their Properties

• A relation $R$ from set $A$ to set $B$ is a subset of the Cartesian product $A \times B$
  • Example: If $A = \{1, 2\}$ and $B = \{a, b\}$, then $R = \{(1, a), (2, b)\}$ is a relation from $A$ to $B$
• Domain of a relation is the set of all first elements (x-coordinates) of the ordered pairs
• Range of a relation is the set of all second elements (y-coordinates) of the ordered pairs
• Reflexive property: A relation $R$ on a set $A$ is reflexive if $(a, a) \in R$ for every $a \in A$
• Symmetric property: A relation $R$ on a set $A$ is symmetric if $(a, b) \in R$ implies $(b, a) \in R$ for all $a, b \in A$
• Transitive property: A relation $R$ on a set $A$ is transitive if $(a, b) \in R$ and $(b, c) \in R$ imply $(a, c) \in R$ for all $a, b, c \in A$
• Equivalence relation is a relation that is reflexive, symmetric, and transitive

Functions as Special Relations

• A function $f$ from set $A$ to set $B$ is a relation where each element in $A$ is paired with exactly one element in $B$
  • Denoted as $f: A \rightarrow B$
• Domain of a function is the set of all inputs (x-values)
• Codomain of a function is the set of all possible outputs (y-values)
• Range of a function is the set of all actual outputs (subset of the codomain)
• One-to-one (injective) function: Each element in the codomain is paired with at most one element in the domain
• Onto (surjective) function: Each element in the codomain is paired with at least one element in the domain
• Bijective function is both one-to-one and onto

Applications in Mathematics

• Set theory is used in various branches of mathematics, including algebra, topology, and analysis
• Venn diagrams visually represent relationships between sets and can be used to solve problems
• Sets and relations are fundamental in defining mathematical structures like groups, rings, and fields
• Functions are essential in calculus, where they model relationships between variables
• Equivalence relations are used to define quotient structures, such as quotient groups and quotient spaces
• Partitions of a set can be defined using equivalence relations
• Set theory is the foundation for mathematical logic and proof techniques

Common Mistakes and How to Avoid Them

• Confusing elements and sets: Remember that elements are objects within a set, while sets are collections of elements
• Misinterpreting set notation: Pay attention to the symbols used, such as $\in$ for "is an element of" and $\subseteq$ for "is a subset of"
• Forgetting to consider the universal set: Always be aware of the context and the universal set when working with complements and subsets
• Misapplying set operations: Be careful when using union, intersection, and difference, and follow the proper order of operations
• Confusing relations and functions: Remember that functions are special types of relations with unique outputs for each input
• Misunderstanding the properties of relations: Carefully check the reflexive, symmetric, and transitive properties when working with relations
• Incorrectly determining the domain and range of functions: Consider all possible inputs and outputs, and identify any restrictions on the domain
• Misusing the terms "one-to-one," "onto," and "bijective": Ensure that you understand the definitions and can apply them correctly to functions