Discrete mathematics離散数学りさんすうがく portal

document_iddoc_df6a61c2ed92005e973b55b377000b5atitle離散数学ポータル講義type講義content_typelecturedate2026-07-01categorymathdescription集合、論理、証明、関係、順序、束、写像をつなげて学ぶ離散数学の入口。

Discrete mathematics離散数学りさんすうがく is mathematics that studies separated objects and their relationships, not quantities that change continuously. The central topics to learn first are sets集合しゅうごう, relations関係かんけい, and maps写像しゃぞう. However, these are not isolated terms. They stand on propositions命題めいだい, predicates述語じゅつご, quantification量化りょうか, and proof証明しょうめい.

Therefore, read this field in the following order.

Use logic論理ろんり and proof証明しょうめい to prepare to read definitions precisely.
Use sets集合しゅうごう and set operations集合演算しゅうごうえんざん to handle collections of objects.
Use mathematical induction数学的帰納法すうがくてききのうほう and recursive definitions to handle natural numbers and recursively built structures.
Use relations関係かんけい to handle connections between objects.
Use orders順序じゅんじょ and lattices束そく to handle comparison and upper-lower structure.
Use maps写像しゃぞう to handle rules that send objects to other objects.

This order expresses dependencies. Later chapters reuse sets集合しゅうごう, relations関係かんけい, and maps写像しゃぞう defined earlier, so do not memorize terms as isolated labels. Follow how the ideas are connected.

1Logic and proof

data/lecture/math/discrete-math/propositions-predicates-and-quantifiers.lecture.n.md data/lecture/math/discrete-math/proof-methods-and-counterexamples.lecture.n.md

2Sets

data/lecture/math/discrete-math/set-basics.lecture.n.md data/lecture/math/discrete-math/set-operations-and-inclusion.lecture.n.md data/lecture/math/discrete-math/families-of-sets-and-index-sets.lecture.n.md data/lecture/math/discrete-math/cartesian-product-basics.lecture.n.md data/lecture/math/discrete-math/power-set-basics.lecture.n.md data/lecture/math/discrete-math/cardinality-and-countability.lecture.n.md

3Induction and recursion

data/lecture/math/discrete-math/mathematical-induction-and-recursive-definitions.lecture.n.md

4Relations

data/lecture/math/discrete-math/relation-basics.lecture.n.md data/lecture/math/discrete-math/relation-composition-and-closure.lecture.n.md data/lecture/math/discrete-math/equivalence-relations-and-partitions.lecture.n.md data/lecture/math/discrete-math/quotient-sets-and-canonical-projections.lecture.n.md

5Orders and lattices

data/lecture/math/discrete-math/partial-and-total-orders.lecture.n.md data/lecture/math/discrete-math/hasse-diagrams-and-maximal-minimal-elements.lecture.n.md data/lecture/math/discrete-math/order-isomorphisms-and-well-orders.lecture.n.md data/lecture/math/discrete-math/lattice-basics.lecture.n.md data/lecture/math/discrete-math/boolean-algebra-basics.lecture.n.md

6Maps

data/lecture/math/discrete-math/map-basics.lecture.n.md data/lecture/math/discrete-math/images-and-preimages-of-maps.lecture.n.md data/lecture/math/discrete-math/injections-surjections-and-bijections.lecture.n.md data/lecture/math/discrete-math/composition-and-inverse-maps.lecture.n.md

7What changes and what does not change

In discrete mathematics離散数学りさんすうがく, each operation asks, "What changes, and what is preserved?"

Object	What changes	What we want to see preserved
Set operation集合演算しゅうごうえんざん	membership conditions for elements	logical structure of membership conditions
Equivalence relation同値関係どうちかんけい	the viewpoint on individual elements	the structure by which elements are classified as the same kind
Order isomorphism順序同型じゅんじょどうけい	names and representations of elements	comparison relations, maximum elements, minimum elements, upper bounds, and lower bounds
Map写像しゃぞう	sending inputs to outputs	the rule, image, inverse image, and type of composition

Having this viewpoint first lets you follow why each definition is needed instead of memorizing definitions by rote.

For each operation, explicitly check which of the input type, membership condition, comparison relation, and reachability is preserved. This check prevents mistaken analogies when similar symbols appear.

8Exercise links

data/exercise/math/discrete-math/logic-and-proof-methods.exercise.n.md data/exercise/math/discrete-math/sets-and-set-operations.exercise.n.md data/exercise/math/discrete-math/cartesian-products-and-power-sets.exercise.n.md data/exercise/math/discrete-math/relations-and-equivalence-relations.exercise.n.md data/exercise/math/discrete-math/order-relations-and-lattices.exercise.n.md data/exercise/math/discrete-math/order-isomorphisms-and-boolean-algebras.exercise.n.md data/exercise/math/discrete-math/maps-injections-and-surjections.exercise.n.md data/exercise/math/discrete-math/images-and-preimages.exercise.n.md

9Summary

At the entrance to discrete mathematics離散数学りさんすうがく, do not memorize sets集合しゅうごう, relations関係かんけい, and maps写像しゃぞう as separate items. Understand them as connected topics built on predicates述語じゅつご, quantification量化りょうか, and proof証明しょうめい.