Mathematical Language and Symbols - Complete Study Guide

Characteristics and Parts of Speech of Mathematical Language

• Definition of Mathematical Language: A formal system used to communicate mathematical ideas clearly, efficiently, and precisely.
• Key Characteristics:
  • Non-temporal: It carries no tense (no past, present, or future context).
  • Emotionless: It carries no emotional content or subjective feeling (it contains no joy or sadness).
  • Precise: It makes statements that are exact and accurate.
  • Concise: It expresses complex mathematical ideas succinctly without unnecessary words.
  • Powerful: It enables complex thoughts and abstract structures to be expressed with relative ease.
• Parts of Speech in Mathematics:
  1. Numbers or Constants: Fixed values (e.g., $1$, $2$, $3$, $\pi$).
  2. Variables: Symbols representing unknown or changeable values (e.g., $x$, $y$, $z$).
  3. Relation Symbols: Symbols indicating a relationship between expressions (e.g., $=$, $>$, $<$).
  4. Operation Symbols: Symbols denoting mathematical operations (e.g., $+$, $-$).
  5. Grouping Symbols: Symbols used to organize expressions and indicate precedence (e.g., parentheses, brackets, braces).

Mathematical Expressions vs. Mathematical Sentences

• Mathematical Expression:
  • Definition: An arrangement of mathematical symbols used to represent a mathematical object of interest.
  • Completeness: Expresses an incomplete thought.
  • Truth Value: Cannot be evaluated as true or false.
  • Examples:
    • $3x + 4y$
    • $x^2$
    • $1 + 1 + 1 + 1$
    • $x + 3$
• Mathematical Sentence:
  • Definition: An arrangement of mathematical symbols that states a complete thought.
  • Completeness: Expresses a complete thought.
  • Truth Value: Can be determined to be (always) true, (always) false, sometimes true, or sometimes false.
  • Examples:
    • $1 + 2 = 3$ (Always True)
    • $1 + 2 > 4$ (Always False)
    • $x = 2$ (Sometimes True / Depends on the value of $x$)
    • $x + 3 = 3 + x$ (Always True)

Conventions in Mathematical Language

• Definition: A mathematical convention is a fact, name, notation, or usage that is universally agreed upon by mathematicians.
• Standard Examples:
  • Order of Operations: The PEMDAS rule (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
  • Triangle Notation: Lowercase letters $a$, $b$, and $c$ denote side lengths; uppercase letters $A$, $B$, and $C$ denote their respective opposite angles.
  • Function Notation: Functions are conventionally denoted using lowercase letters such as $f$, $g$, or $h$.
  • Coefficient Placement: Numerical coefficients are always written before variables (e.g., $3x$ instead of $x3$).
  • Angle Representation: Lowercase Greek letters such as $\alpha$ (alpha), $\beta$ (beta), and $\theta$ (theta) represent angle measures.
  • Summation Operator: The capital Greek letter sigma $\Sigma$ represents summation.

Fundamentals of Set Theory

• Definitions:
  • Set: Any group or collection of distinct objects.
  • Elements: The individual objects belonging to a set.
• Set Membership Notation:
  • If an object $a$ is an element of set $A$, it is written as $a \in A$.
  • If an object $a$ is not an element of set $A$, it is written as $a \notin A$.
  • Examples:
    • $4 \in \{2, 3, 4, 7\}$ is a true statement.
    • $-5 \in \{2, 3, 4, 7\}$ is a false statement.
• Basic Number Sets:
  • Natural Numbers / Counting Numbers ($\mathbb{N}$): $\mathbb{N} = \{1, 2, 3, 4, 5, \dots\}$
  • Integers ($\mathbb{Z}$): $\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$
  • Rational Numbers ($\mathbb{Q}$): Numbers that can be written in the quotient form $\frac{m}{n}$ where $m, n \in \mathbb{Z}$ and $n \neq 0$.
  • Irrational Numbers ($\mathbb{Q}^C$): Real numbers that cannot be expressed as a ratio of two integers.
  • Real Numbers ($\mathbb{R}$): The comprehensive set containing all rational and irrational numbers.
• Methods of Describing Sets:
  • Listing Method (Roster Method): All or partial elements of the set are explicitly listed inside set braces.
  • Set-Builder Method: Elements are defined by specifying the properties that elements must satisfy to belong to the set.
  • Comparative Examples:
    • Set $A$ (Natural numbers less than $10$):
      • Listing Method: $A = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$
      • Set-Builder Method: $A = \{x \mid x \in \mathbb{N}; x < 10\}$
    • Set $B$ (Even integers between $4$ and $12$):
      • Listing Method: $B = \{6, 8, 10\}$
      • Set-Builder Method: $B = \{x \mid x \in \mathbb{Z}; x = 2n, n \in \mathbb{Z}; 4 < x < 12\}$
    • Set $C$ (Odd integers):
      • Listing Method: $C = \{1, 3, 5, 7, 9, \dots\}$
      • Set-Builder Method: $C = \{x \mid x \in \mathbb{Z}; x = 2n + 1, n \in \mathbb{Z}\}$
• Set Classifications and Terminology:
  • Empty Set / Null Set: A set containing no elements, denoted by $\emptyset$ or {}.
  • Unit Set: A set containing exactly one element.
  • Finite Set: A set with a countable or limited number of elements.
  • Infinite Set: A set with an unlimited number of elements.
  • Cardinal Number: The total number of elements contained in a finite set.
  • Equal Sets ($A = B$): Set $A$ is equal to set $B$ if and only if they contain exactly the same elements.
  • Equivalent Sets ($A \sim B$): Set $A$ is equivalent to set $B$ if and only if they contain the same number of elements (equal cardinality).
  • Universal Set ($U$): The master set containing all potential elements under consideration.

Set Operations

• Complement of a Set:
  • Definition: The complement of set $A$ (denoted $A^C$) contains all elements of the universal set $U$ that are not elements of A$.\n * **Example:** If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$and$S = {2, 4, 6, 7}$, then$S^C = {1, 3, 5, 8, 9, 10}.\n* **Subsets and Proper Subsets:**\n * **Subset (A \subseteq B$):** Set$A$is a subset of set$B$if and only if every element of$A$is also an element of$B$. If$A$is not a subset of$B$, it is written as$A \nsubseteq B.\n * **General Laws:** For any arbitrary set A$,$A \subseteq A$(every set is a subset of itself) and$\emptyset \subseteq A (the empty set is a subset of every set).\n * **Evaluated Subset Statements:**\n * {5, 10, 15, 20} \subseteq {10, 15, 20, 25, 30} (False)\n * \mathbb{Z} \subseteq \mathbb{Q} (True)\n * {2, 4, 6} \subseteq {2, 4, 6} (True)\n * **Proper Subset (A \subset B$):** Set$A$is a proper subset of set$B$if and only if every element of$A$is an element of$B$, and$A \neq B.\n* **Union of Sets:**\n * **Definition:** The union of sets A$and$B$(denoted$A \cup B$) is the set of all elements belonging to$A$, to$B, or to both.\n * **Set-Builder Form:** A \cup B = {x \mid x \in A \text{ or } x \in B}\n * **Examples:** Let A = {1, 4, 5, 7}$,$B = {2, 3, 4, 5, 6}$, and$C = {3, 6, 9}.\n * A \cup B = {1, 2, 3, 4, 5, 6, 7}\n * A \cup C = {1, 3, 4, 5, 6, 7, 9}\n* **Intersection of Sets:**\n * **Definition:** The intersection of sets A$and$B$(denoted$A \cap B$) is the set containing all elements belonging simultaneously to both$A$and$B$.
  • Set-Builder Form: $A \cap B = \{x \mid x \in A \text{ and } x \in B\}$
  • Examples: Let $A = \{1, 4, 5, 7\}$, $B = \{2, 3, 4, 5, 6\}$, and $C = \{3, 6, 9\}$.
    • $A \cap B = \{4, 5\}$
    • $A \cap C = \emptyset$
• Disjoint Sets:
  • Definition: Two sets are disjoint if their intersection yields an empty set ($A \cap B = \emptyset$).
  • Example: Given $A = \{1, 4, 5, 7\}$ and $C = \{3, 6, 9\}$, $A \cap C = \emptyset$; therefore, $A$ and $C$ are disjoint sets.

Functions and Relations

• Ordered Pairs: Pairs of elements written in a specific order as $(x, y)$, where $x$ is the input (domain element) and $y$ is the output (range element).
• Relation: A collection of ordered pairs $(x, y)$ establishing a connection between elements of two sets, mapping inputs to outputs.
• Function: A function $F$ from domain set $A$ to co-domain set $B$ is a specific relation satisfying two mandatory conditions:
  1. For every element $x \in A$, there exists an element $y \in B$ such that (x, y) \in F$.\n 2. For all x \in A$and$y, z \in B$, if$(x, y) \in F$and$(x, z) \in F$, then$y = z (no input maps to multiple distinct outputs).\n* **Classification Examples of Sets of Ordered Pairs:**\n * {(a, c), (b, d)}: Classified as a **Function**.\n * {(a, c), (a, d), (b, e)}$: Classified as a **Relation** (not a function, because input$a$maps to both$c$and$d).\n * {(a, d), (b, d), (c, d)}: Classified as a **Function**.\n * {(a, c), (a, d), (b, d), (b, e), (c, d)}: Classified as a **Relation** (not a function).\n\n# Binary Operations\n\n* **Definition:** A binary operation is a rule combining two elements from a set to produce a single new element of that set. Standard operations like addition or multiplication performed on elements of a set are primary examples.\n* **Notation:** Represented by an operator symbol such as *$performed on a set$A$.
• Closure Property: A non-empty set $A$ is closed under a binary operation $*$ if for all $a \in A$ and $b \in A$, the operation yields a * b \in A$.\n* **Examples of Closure:**\n * a * b = ab + a + b$for all$a, b \in \mathbb{Z}: **Closed binary operation** (the product and sum of integers are always integers).\n * a * b = \frac{a}{b}$for all$a, b \in \mathbb{Z}$: **Not closed** (e.g.,$\frac{1}{2} \notin \mathbb{Z}$, and division by zero is undefined in$\mathbb{Z}).\n\n# Mathematical Logic and Statements\n\n* **Logic:** The formal study of principles and methods used to distinguish valid from invalid reasoning.\n* **Statement:** A declarative sentence that is definitively either true or false, but not both.\n * *Declarative Statement Example:* "Florida is a state in the United States." (Valid Statement - True).\n * *Non-Statement Example 1:* "How are you?" (Interrogative sentence; cannot be true or false).\n * *Non-Statement Example 2:* x + 1 = 5$(Open sentence; truth value depends on unassigned variable$x).\n* **Types of Statements:**\n * **Simple Statement:** A statement conveying a single independent idea.\n * **Compound Statement:** A statement combining two or more simple statements using logical connectives.\n* **Logical Variables and Connectives:** Letters such as p$,$q$, and$r symbolize simple statements, which are joined by logical connectives ("and", "or", "if...then", "if and only if").\n\n# Compound Statements and Truth Rules\n\n* **Negation:**\n * **Statement Form:** Not p\n * **Symbolic Form:** \sim p\n * **Truth Rule:** Carries the exact opposite truth value of p\n * **Example:**\n * p: Missy has a dog.\n * \sim p: Missy does not have a dog.\n* **Conjunction:**\n * **Statement Form:** p$and$q\n * **Symbolic Form:** p \land q\n * **Truth Rule:** True only if both p$and$q are true.\n * **Example:**\n * p$:$5 is a counting number (True).\n * q$:$5 is an even number (False).\n * p \land q$:$5$is a counting number and$5 is an even number (Truth value: **False**).\n* **Disjunction:**\n * **Statement Form:** p$or$q\n * **Symbolic Form:** p \lor q\n * **Truth Rule:** True if at least one component statement is true.\n * **Example:**\n * p \lor q$:$5$is a counting number or$5 is an even number (Truth value: **True**).\n* **Conditional:**\n * **Statement Form:** If p$, then$q\n * **Symbolic Form:** p \rightarrow q\n * **Terminology:** Statement p$is the **antecedent**; statement$q is the **consequent**.\n * **Truth Rule:** False only when the antecedent p$is true and the consequent$q is false; true in all other cases.\n * **Examples:**\n * p$:$2$is an integer,$q$:$2$is a rational number.$p \rightarrow q$: "If$2$is an integer, then$2 is a rational number." (Truth value: **True**).\n * "If 3$is a negative number, then$5 > 7" (Antecedent false, consequent false; overall statement: **True**).\n * "If 5 > 3$, then$2 + 7 = 4" (Antecedent true, consequent false; overall statement: **False**).\n* **Biconditional:**\n * **Statement Form:** p$if and only if$q\n * **Symbolic Form:** p \leftrightarrow q\n * **Truth Rule:** True if both p$and$q share the identical truth value (both true or both false).\n * **Examples:**\n * p$:$x + 4 = 7$,$q$:$x = 3$.$p \leftrightarrow q$: "$x + 4 = 7$if and only if$x = 3".\n * r$:$x^2 = 36$,$s$:$x = 6$.$r \leftrightarrow s$: "$x^2 = 36$if and only if$x = 6".\n\n# Quantifiers in Mathematical Logic\n\n* **Existential Quantifiers:**\n * **Definition:** Words or phrases used as prefixes to assert the existence of at least one element satisfying a property.\n * **Keywords:** "Some", "there exists", "at least one".\n * **Example:** "There exists a real number whose multiplicative inverse is itself."\n* **Universal Quantifiers:**\n * **Definition:** Words used to express that every element of a given set satisfies a condition, or to deny existence entirely.\n * **Keywords:** "None", "no", "all", "every".\n * **Denial of Existence:** "None" and "no".\n * **Universal Assertion:** "All" and "every".\n * **Example:** "All elements of the set of integers are elements of the set of rationals."\n\n# Related Conditional Statements\n\n* **Variants of Conditional Statements:**\n * **Conditional Statement:** p \rightarrow q$("If$p$, then$q")\n * **Converse:** q \rightarrow p$("If$q$, then$p")\n * **Inverse:** \sim p \rightarrow \sim q$("If not$p$, then not$q")\n * **Contrapositive:** \sim q \rightarrow \sim p$("If not$q$, then not$p")\n* **Verbal Example:**\n * **Conditional (p \rightarrow q):** If I get the job, then I will rent the apartment.\n * **Converse (q \rightarrow p):** If I rent the apartment, then I will get the job.\n * **Inverse (\sim p \rightarrow \sim q):** If I don't get the job, then I will not rent the apartment.\n * **Contrapositive (\sim q \rightarrow \sim p):** If I don't rent the apartment, then I will not get the job.\n* **Mathematical Example:**\n * **Conditional (p \rightarrow q$):** If a number is divisible by$4, then it is even.\n * **Converse (q \rightarrow p$):** If a number is even, then it is divisible by$4$.
  • Inverse ($\sim p \rightarrow \sim q$): If a number is not divisible by $4$, then it is not even.
  • Contrapositive ($\sim q \rightarrow \sim p$): If a number is not even, then it is not divisible by $4$.