Mathematics 101
- Reminder of Fundamentals
- Advanced Arithmetics
- Introduction to Geometry
- In-depth Geometry
- Deeper into Numbers
- Further into Algebra
- Elementary Statistics & Probability
- Advanced Statistics, Probability
- Mathematical Logic
- Calculus
- Calculus
- Trigonometry I
- Trigonometry II & Conclusion
Mathematical Logic
Introduction to Mathematical Logic
The study of mathematics itself using mathematical methods.
Mathematical logic is a subfield of mathematics that explores the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science.
Importance of Logic in Mathematics
Logic is the backbone of mathematical reasoning. It provides a systematic approach to proving mathematical theorems and solving problems. It helps us understand the structure of mathematical arguments and ensures the correctness of mathematical proofs.
Basic Logical Operations
There are three basic logical operations: AND, OR, and NOT.
- AND (Conjunction): The AND operation between two statements is true only if both statements are true.
- OR (Disjunction): The OR operation between two statements is true if at least one of the statements is true.
- NOT (Negation): The NOT operation negates the truth value of a statement. If a statement is true, its negation is false, and vice versa.
Truth Tables and Logical Equivalences
A truth table is a mathematical table used in logic to compute the functional values of logical expressions on each of their functional arguments. It is used to determine the truth value of a compound statement based on the truth values of its simple components.
Logical equivalences are statements that are structurally different but have the same truth value. For example, the statement "P AND Q" is logically equivalent to "Q AND P".
Conditional Statements and Logical Implications
A conditional statement, symbolized by P -> Q, is an if-then statement in which P is a hypothesis and Q is a conclusion. The logical connector in a conditional statement is denoted by the symbol ->.
Logical implication is a logical operation that takes two logical statements P and Q and produces a new statement of the form "If P then Q". It is true except for the case where P is true and Q is false.
Quantifiers: Universal and Existential
Quantifiers are used in mathematics to specify the scope of a statement.
- Universal Quantifier (∀): The universal quantifier "for all" expresses that a statement is true for all elements in a certain set. For example, "For all x in the set of natural numbers, x is greater than or equal to 0".
- Existential Quantifier (∃): The existential quantifier "there exists" expresses that there is at least one element in a certain set for which a statement is true. For example, "There exists an x in the set of natural numbers such that x is less than 5".
Understanding mathematical logic is crucial for anyone studying mathematics or related fields. It provides the foundation for constructing mathematical proofs and reasoning about mathematical statements.