Discrete Mathematics/Structures (Basics)

Explore the core topics in discrete mathematics and structures for beginners, outlining the essential concepts covered in this basics course.

Define sets as ordered or unordered collections of elements, represented by curly brackets, with roaster and builder forms explained, and distinguish listing, property-based, and descriptive definitions.

Learn standard notations in discrete mathematics for sets, including element of, subset, proper subset, union, intersection, difference, and complement. These symbols describe relationships and operations on sets.

Explore the types of sets, including equal, equivalent, and universal sets, plus empty, singleton, finite, infinite, and disjoint sets. Learn about subsets, proper and improper subsets, and the power set.

Explore the operations of sets, including union, intersection, and the difference, plus the complement of a set, with examples.

Explore the algebra of sets by applying nine laws: associative, commutative, distributive, identity, complements, double complement, and dominance, through unions and intersections with proofs.

Explore the algebra of sets (part 2) focusing on distributive and De Morgan laws for union and intersection of three sets, with proofs and identity and universal set concepts.

This lecture on algebra of sets covers complement, negation, union and intersection with the universal set and empty set, proving double complement and dominance laws.

Explore Venn diagrams by representing the universal set as a rectangle and each set as circles, showing disjoint and non-disjoint cases, along with union, intersection, difference, and subset relationships.

Summarizes the topics covered in discrete mathematics and structures basics, providing a concise overview of the foundational concepts for this course.

Explore disjoint sets, their cardinality, and the union principle, including the inclusion-exclusion idea; distinguish finite from infinite cardinalities through simple examples.

Explore the inclusion-exclusion principle for non-disjoint sets, derive the union formula |A∪B| = |A| + |B| − |A∩B|, and apply it to language and programming examples.

Apply the three-set inclusion-exclusion principle to compute unions and overlaps among three sets, illustrated with course enrollments (economics, English, mathematics) and three-game viewing.

This lecture explains the inclusion-exclusion principle for finitely many sets, showing how to compute the number of students who took at least one course using unions and intersections.

Explore the topics covered in discrete mathematics basics, emphasizing foundational structures and key concepts from the course. Build a solid understanding of how these elements interact within discrete systems.

Learn the principle of mathematical induction, including base step and inductive step, proving statements for all natural numbers, with illustrations of tiles falling and the smallest inductive set.

Explore mathematical induction through examples, establishing base and induction steps to prove identities like the sum of squares equals n(n+1)(2n+1)/6 and related formulas for natural numbers.

Explore mathematical induction through concrete examples, proving that 7^n minus 3^n is divisible by four and that 2^n exceeds n for all natural numbers, via base and inductive steps.

Topics covered in discrete mathematics and structures basics guide learners through essential concepts and foundational topics in this course.

Apply basic counting principles for permutation and combination, including product and addition rules for independent and disjoint tasks, to compute totals like 57 and 40.

Apply the product rule from the basic counting principle to combine tasks, such as labeling seats with 26 letters and 100 numbers, 2600 ways, and 26^3 × 10^3 license plates.

Learn the subtraction rule, also known as inclusion-exclusion, for counting unions of two sets using |E1 ∪ E2| = |E1| + |E2| − |E1 ∩ E2|, with box-and-object examples.

Explore permutations by arranging objects in a given order using nPr = n!/(n-r)!, with examples like 5P3 = 60, and cover restrictions, repetitions, and circular permutations.

Study permutation with concrete examples—arranging letters into words, selecting eight from nine without repetition (9P8), grouping books, creating repeating four-digit codes, and circular arrangements.

Explore combinations where order does not matter and learn to compute n choose r using the factorial formula n!/(r!(n−r)!), noting edge cases r=0 or r=n.

Learn how combinations work through practical examples, such as choosing five zeros from sixteen, three from ten, and four from seven, using nCr and factorials.

Explore the difference between permutation and combination, showing how order matters in permutations and not in combinations. Learn the formulas nPr = n!/(n−r)!, nCr = n!/(r!(n−r)!).

Explore topics covered in discrete mathematics and structures basics, outlining foundational concepts and key areas for beginners.

Explore the formal logic of propositional calculus, defining statements and propositions, and build truth tables to evaluate compound propositions using proposition variables and logical connections.

Construct truth tables for two propositions, B and Q, using negation and if and only if; draw the table with all four combinations and determine the final target value.

Explain the combination of propositions in proposition calculus, showing how multiple propositions yield a proposition, with true and false denoted by B and F, including fundamental and design connectors.

Explore fundamental connectors in discrete structures, focusing on conjunction, disjunction, and negation with truth tables and examples using propositions B and Q.

Explore derived connectors: nand gate, nor gate, and xor, with their negations and basic truth patterns for P and Q.

Explore conditional statements and logical connectors, including implication, converse, inverse, contrapositive, and biconditional (if and only if), with examples and truth conditions.

Explore propositional logic basics by analyzing conjunction, disjunction, and negation with symbolic forms and truth tables, and apply implication and XOR to real-world examples.

Explore truth tables for propositions using and, or, not; cover B and not B, B or not B, and P, Q, R, with eight outcomes.

Learn the laws of algebra for propositions, including domination, identity, double negation, and negation. Apply commutative, associative, distributive, and absorption laws to or and and operations.

Analyze the definitions and truth values of propositions, covering tautology, contradiction, contingency, negation, conjunction, disjunction, and logical equivalence for clear reasoning.

Explore the basics of discrete mathematics and structures through the topics covered in this course, offering a clear overview of fundamental concepts.

Explore the basics of sets vs. ordered tuples in discrete structures: how order and repetition define equality, from two-element ordered pairs to longer sequences and their components.

Describe the cartesian product of sets, forming ordered pairs from a and b, note that a×b is not equal to b×a, and extend to three or more sets as tuples.

Discusses Cartesian product and its properties, including non competitive (non commutative) behavior, equality conditions, associativity with A×B, distributive laws over intersection and union, and cardinality concepts.

Explore the concept of relations between sets, defined as subsets of the Cartesian product, with focus on binary relations, domain, and range, including unity relations and examples.

Explore the types of relations in discrete structures, including empty, universal, identity, and inverse relations, and examine reflexive relations using Cartesian products.

Explore six types of relations, including symmetric, anti-symmetric, reflexive, transitive, and equivalence relations, and distinguish partial orders from other relation classes with illustrative examples.

Explore the composition of relations and the completion of relations, linking A to B and B to C, and examine identity relations and R S notation through examples.

Explain the composition of relations with a concrete example. Then determine the completion and inverse of two relations R1 and R2, illustrating the process with labeled element sets.

Define the closure properties of relations by adding minimum elements to achieve reflexive, symmetric, and transitive closures, with examples illustrating how to extend relations to satisfy each property.

Explain the closure properties of a relation on a set, including reflexive, symmetric, and transitive closures, and how union, inverse, and composition form the smallest closures.

Shows how relations between finite sets are captured by a matrix, with rows as A and columns as B, using 1s and 0s, exploring reflexive, symmetric, antisymmetric properties.

Learn how to represent relations with directed graphs, using vertices and edges (arcs), including self-loops, and examine reflexive, asymmetric, and transitive properties with examples.

Examine directed graphs through three examples to see reflexive, symmetric, anti-symmetric, and transitive properties in relation diagrams.

Compare linear (total) orders and partial orders on a set using transitivity and the law of dichotomy, distinguishing comparable from non-comparable elements. Grasp lexicographic order and identity relations in ordering.

Explore the basics of discrete mathematics and structures through a survey of fundamental topics, preparing learners with essential concepts for further study.

Define functions as mappings from A to B with one output per input; illustrate with graphs, diagrams, and the 3 maps to 6 example.

Defines a function as a total, single-valued relation from A to B, and introduces domain, range, image, and preimage with examples of inverses and unique outputs.

Learn recursively defined functions in discrete mathematics, where f(n) relies on prior values via a recurrence such as f(n)=3 f(n-1)+2. The lecture shows deriving successive terms from values and recurrences.

Explore one-to-one (injective) functions where every domain element has a unique image, and onto functions where every codomain element is represented; some functions are both.

Explore other types of functions beyond basics, using coordinates to represent functions from real numbers. Identify identity, constant, and rational functions, and examine their graphs, domains, and ranges.

Examine polynomial functions by degree, from constant and linear to quadratic and cubic, using standard forms such as f(x)=a0+a1 x+a2 x^2 and f(x)=a x^3+b x^2+c x+d.

Explore other types of functions in discrete mathematics, including the modulus (absolute value) function and the step function, with definitions and basic graphing insights.

Explore addition, subtraction, multiplication, and division of functions f and g, demonstrating f+g, f-g, f*g, and f/g with x-based examples.

Learn how to find inverse functions by verifying bijectivity (one-to-one and onto) and deriving the inverse mapping from Y back to X through algebraic solving.

defines equal functions as f and g from X to Y where f(x) = g(x) for every x in the domain. includes examples solving for x and noting non-equality.

Master the composition of functions by applying g to x then f to the result, forming f∘g from A to C. Discover how one-to-one and onto properties carry through compositions.

Explore how many functions can be formed from a finite set to another, using factorial to count permutations and illustrating domain, codomain, and function mapping with examples.