Master logic, set theory, combinatorics, and mathematical structures
← Back to MathematicsMaster the foundations of mathematical logic and logical reasoning.
Extend logic to work with predicates, variables, and quantifiers.
Learn the fundamental concepts of sets and set operations.
Study mathematical functions, relations, and their properties.
Master various methods of mathematical proof and reasoning.
Explore properties of integers and fundamental number theory concepts.
Learn counting techniques and combinatorial reasoning.
Study graphs, networks, and their mathematical properties.
Solve recurrence relations and analyze recursive algorithms.
Learn Boolean algebra and its applications to digital logic.
Study abstract algebraic structures and their properties.
Apply discrete mathematics to computer science and real-world problems.
Build the foundation of mathematical reasoning with propositional logic.
Understand what makes a statement a proposition: declarative sentences that are either true or false. Learn to identify propositions vs. questions, commands, or ambiguous statements.
Master the five main logical connectives: conjunction (∧), disjunction (∨), negation (¬), implication (→), and biconditional (↔). Understand their meaning and usage.
Construct and interpret truth tables for compound propositions. Use truth tables to determine when complex logical statements are true or false.
Learn important logical equivalences like De Morgan's laws, distributive laws, and absorption laws. Use these to simplify complex logical expressions.
Identify tautologies (always true statements) and contradictions (always false statements). Understand their role in logical reasoning and proof.
Master conditional statements (if-then) and biconditionals (if and only if). Understand the difference between converse, inverse, and contrapositive.
Analyze the validity of logical arguments. Distinguish between valid and invalid arguments, and between validity and soundness.
Learn fundamental rules of inference like modus ponens, modus tollens, hypothetical syllogism, and disjunctive syllogism for constructing proofs.
Extend logic to handle variables, predicates, and quantified statements.
Understand predicates as functions that return true or false values. Learn to work with predicates involving one or more variables.
Master the universal quantifier (∀x) meaning "for all x". Learn to read, write, and interpret universally quantified statements.
Understand the existential quantifier (∃x) meaning "there exists an x". Practice constructing and interpreting existentially quantified statements.
Work with multiple quantifiers in the same statement. Understand the importance of quantifier order and how switching order changes meaning.
Learn to negate quantified statements correctly: ¬(∀x P(x)) ≡ ∃x ¬P(x) and ¬(∃x P(x)) ≡ ∀x ¬P(x). Apply De Morgan's laws to quantifiers.
Understand how the domain of discourse affects the truth value of quantified statements. Work with restricted domains and implicit quantification.
Analyze arguments involving quantified statements. Use rules of inference extended to predicate logic, including universal and existential instantiation.
Learn equivalences involving quantifiers and logical connectives. Understand when quantifiers can be distributed over conjunctions and disjunctions.
Master the fundamental language of mathematics through set theory.
Learn set notation including roster and set-builder notation. Understand elements, membership (∈), empty set (∅), and subset relationships (⊆).
Master fundamental set operations: union (∪), intersection (∩), complement, and difference. Understand their properties and relationships.
Use Venn diagrams to visualize sets and set operations. Apply them to solve problems involving multiple sets and their relationships.
Learn important set identities including commutative, associative, distributive, De Morgan's laws, and absorption laws for sets.
Understand Cartesian products A × B as sets of ordered pairs. Learn properties of Cartesian products and their relationship to relations and functions.
Master the concept of power sets P(A) containing all subsets of A. Understand the relationship between |A| and |P(A)| = 2^|A|.
Learn to count elements in finite sets using inclusion-exclusion principle. Understand cardinality of infinite sets and countability concepts.
Understand Russell's paradox and why naive set theory needs axioms. Learn basic axioms of set theory and their importance in mathematics.
Study the fundamental concepts of mathematical functions and relations.
Understand relations as subsets of Cartesian products and functions as special relations. Learn the formal definition of functions and their notation.
Distinguish between domain (input set), codomain (target set), and range (actual output set). Understand how these concepts define a function completely.
Master one-to-one (injective) functions where different inputs produce different outputs. Learn tests for injectivity and their applications.
Understand onto (surjective) functions where every element in the codomain is mapped to. Learn to prove and disprove surjectivity.
Study bijective functions that are both injective and surjective. Understand their importance in establishing one-to-one correspondence between sets.