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Example Sentences

Example sentences are the most effective way to learn both grammar and vocabulary simultaneously. A single well-chosen sentence teaches word meaning, grammatical context, tonal environment, and natural usage all at once. Grammar rules are generalizations — sentences are the data. How to Use This Section By grammar pattern. Each major pattern has a dedicated page with 20–50 examples at increasing complexity. Use these when you encounter a pattern you do...

Maths › 00. General Mathematics ›

06.5 Recursion and Induction

6.5 Recursion and Induction Recursion and induction are paired patterns. Recursion defines objects step by step. Induction proves statements about objects built step by step. The natural numbers give the basic model. Start with $0$. Then repeatedly apply the successor operation: $$ 0,\quad 1,\quad 2,\quad 3,\quad \dots $$ Each number is either the starting value or obtained from an earlier number. This recursive construction supports the principle of induction. To...

Maths › 00. General Mathematics ›

06.2 Symmetry and Invariance

6.2 Symmetry and Invariance Symmetry is the presence of transformations that preserve structure. An object has symmetry when it can be changed in some way while remaining the same for the purpose being studied. A square can be rotated or reflected and still remain a square. A graph can have its vertices renamed while preserving adjacency. A vector space can be described using different bases while preserving its linear structure....

Maths › 00. General Mathematics ›

06.3 Local-to-Global

6.3 Local-to-Global Principles A local-to-global principle studies an object by examining small parts and then asking whether those local facts determine the whole. This pattern appears whenever a large structure is too complex to understand directly. Local information is information available near a point, on a small region, or on a restricted part of an object. Global information concerns the entire object. For example, a curve may look like a...

Maths › 00. General Mathematics ›

06.4 Decomposition and Composition

6.4 Decomposition and Composition Decomposition and composition are inverse patterns. Decomposition breaks a complex object into simpler parts. Composition builds a complex object from simpler parts. Much of mathematics moves between these two directions. Decomposition is useful because large objects are often hard to understand directly. A problem becomes easier when its pieces can be studied separately. Composition is useful because it explains how local or partial information forms a...

Maths › 00. General Mathematics ›

06. Patterns Across Mathematics

Chapter 6. Patterns Across Mathematics Mathematics is organized by recurring patterns. The same forms of reasoning appear in algebra, geometry, topology, analysis, combinatorics, computation, and logic. Once these patterns are recognized, new subjects become easier to learn because their methods are no longer isolated. This chapter studies several of the most common patterns. Duality appears when a concept has a counterpart obtained by reversing direction, exchanging roles, or changing perspective....

Maths › 00. General Mathematics ›

06.1 Duality

6.1 Duality Duality is a recurring pattern in mathematics where one concept is paired with another by reversing roles, directions, or operations. A statement about one object often has a corresponding statement about its dual, obtained by systematically exchanging parts of the structure. At a simple level, duality appears in operations. Intersection and union of sets behave as opposites under complement: $$ (A \cup B)^c = A^c \cap B^c, \quad...

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Example Sentences

06.5 Recursion and Induction

06.2 Symmetry and Invariance

06.3 Local-to-Global

06.4 Decomposition and Composition

06. Patterns Across Mathematics

06.1 Duality