Master Discrete Mathematics: Sets, Math Logic, and More

Explore propositional logic by identifying propositions with truth values, using connectives (not, and, or, exclusive or, implication, biconditional), translating statements between English and logic, and building truth tables.

Explore how truth tables in discrete mathematics verify propositions built from connectives like negation, conjunction, disjunction, exclusive or, and implication, including biconditional cases.

Master discrete mathematics using truth tables to prove logical equivalence, tautologies, and Morgan's laws, by constructing step-by-step comparisons of P, Q, not P, and not Q.

Practice with truth tables to analyze all possible combinations of P and Q, including not P, not Q, P and Q, and implications, culminating in tautology identification.

Analyze truth tables to identify tautologies and logical equivalences. Use P, Q, and not P and not Q to assess P or Q and P implies Q.

Master the basic laws of logic to simplify complex logical expressions. Learn and apply double negation, identity and domination laws, tautology, contradiction, and De Morgan's laws.

Explore advanced laws of logic to simplify complex logical expressions, including tautologies, contradictions, distributive, commutative, and associative properties. Learn absorption, inverse, and conditional laws with truth-table proofs and practical examples.

Practice with logic laws to prove equivalences like p implies q equals not q implies not p, using conditional, double negation, and commutative properties.

Explore a walkthrough of logic laws quiz, evaluating negation, contradictions, tautology, and implication. Think about it before truth tables and see that not P or Q equals P implies Q.

Discover the sum rule of counting and the product principle, and learn to apply them to counting problems. See examples with cards, phones, and plates to count possibilities.

Apply the sum rule and product rule to counting problems, including license plates with odd numbers and vowels, race placements, and three-flavor cones with order and no repetition.

Master factorials by understanding n! as n×(n−1)×...×1, with 0! = 1, and use cancellation in ratios like 7!/2! to simplify probability calculations, including combinations and permutations.

Apply the product rule and permutation concepts to count options in painting designs, seating arrangements, travel routes, and passwords with four numbers and four letters.

Master permutations with and without repetition using the product principle and factorial notation, illustrated by examples like phone passcodes, top-three race finishes, and P(n,r) arrangements such as Mississippi.

Explore how combinations ignore order, distinguishing with and without repetition. Learn formulas for selecting items, from lottery and poker hands to ice cream scoops, with and without repetition.

Master permutations and combinations by distinguishing when order matters and when it does not, including repetition and without repetition cases. Apply core formulas and practice problems to count arrangements.

Explore harder practice with permutations and combinations, including circular-table arrangements and factorials, apply n choose r, and analyze probability of all-same-color hands.

Explore permutations and factorials by counting arrangements with no repetition, such as four questions with four choices and seven switches with up or down. Learn how factorials reduce to n(n-1).

Explore combination quiz techniques, including seven choose four, twelve choose five, deck of cards with queens, and applying factorials to count valid hands.

Explore the binomial theorem, its binomial coefficients, and how to expand expressions like s plus y using summations and combinations. See how Pascal's triangle and induction underpin the pattern.

Explore how Pascal's triangle generates binomial coefficients and expand X plus Y to the fourth using the triangle, highlighting faster computation and coefficient patterns.

Walk through the binomial theorem to find the fourth term of (2x−y)^5 and its coefficient, using n choose k calculations. Compare this approach with Pascal's triangle to identify terms.

Explore the basics of proofs, including definitions, theorems, and methods like direct proofs, contrapositive, contradiction, cases, and induction, with parity and divisibility.

Apply direct proofs by assuming P is true and deriving Q, illustrated through odd n implying n^2 is odd, and divisibility and differences of squares.

Master discrete mathematics uses contrapositive reasoning to prove P implies Q by showing not Q implies not P, with parity examples over integers.

Master proof by contradiction: assume the negation of a claim and derive a contradiction, illustrated with the irrationality of the square root of two. Compare direct, contrapositive, and contradiction proofs.

Apply proof by cases to prove P implies Q by splitting into even and odd cases, and show parity: X and Y have same parity iff X+Y is even.

Learn a proof by cases in discrete mathematics: show that a non-multiple of three has n squared equals three k plus one, via n = 3m+1 or n = 3m+2.

Explore mathematical induction: establish a base case, prove an inductive step, and conclude for all n, with domino proof, sum of odd numbers equals n^2, and n^3+2n divisible by 3.

Learn the basics of graph theory by defining graphs as a collection of vertices and edges, and distinguish undirected and directed graphs, incidents, adjacency, loops, isolated vertices, and parallel edges.

Explore common graph types, including simple graphs with no loops or parallel edges, complete graphs, bipartite graphs, and complete bipartite graphs such as K4 or K2,3.

Explore degrees of graphs by counting incident edges, noting loops count twice, and learn that in complete graphs each vertex has degree n-1 and total degree n(n-1).

Explore paths and circuits on graphs, distinguishing paths from circuits and identifying simple versus non-simple variants. Learn how connectedness, edge repetition, and vertex repetition define graph traversal.