Mathematics / foundation
Sets, relations, and finite counting
Build the discrete language of mathematics through maps, equivalence classes, cardinality, and systematic counting arguments.
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Module 01 foundation
Sets, relations, and finite counting
Build the discrete language of mathematics through maps, equivalence classes, cardinality, and systematic counting arguments.
- 01
Set operations: turn membership into an argument
Distinguish elements from subsets and prove identities by following one arbitrary member.
21 min - 02
Cartesian products and relations: record which pairs are allowed
Build ordered pairs, interpret a relation as a subset of a product, and compose relations carefully.
22 min - 03
Injections, surjections, and bijections: what an inverse needs
Track collisions and missing outputs, construct inverse maps, and compare finite sizes through functions.
23 min - 04
Equivalence classes: group objects without making the rule ambiguous
Derive partitions from equivalence relations and test whether a rule on classes is well-defined.
24 min - 05
Partial orders: compare what can be compared
Separate antisymmetry from symmetry, draw cover relations, and distinguish maximal from greatest elements.
24 min - 06
Finite cardinality: force a collision without locating it
Use injections and occupancy bounds to prove that some category must contain several objects.
23 min - 07
Sums, products, and double counting: name the objects first
Count disjoint alternatives, sequential choices, and incidences without hiding overlap or multiplicity.
25 min - 08
Permutations and combinations: decide whether order survives
Derive factorial formulas from reversible constructions and explain binomial identities by subsets.
25 min - 09
Inclusion–exclusion: repair overlap systematically
Derive alternating intersection counts and use them to count permutations with no fixed points.
27 min - 10
Countability and diagonal arguments: when no list can suffice
Enumerate integers, pairs, and rationals, then construct an object missing from any proposed list of subsets.
28 min