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In mathematics as in daily life concrete observations that repeat, are described once and then referred to later. The common pattern in many observations is called abstrakt. They make descriptions shorter in their totality.

For learners it is often difficult to link the abstract with the concrete again. Therefore it is good to comment definitions and theorems with motivation, examples and applications.

First one normally starts with the concrete, recognizes common patterns by comparing, which is called analysis, and the result is the abstraction. This is reduction of redundancy and compression, a fundamental cognitive process.

Note

Mathematics often uses the equivalence relation (= reflexive, symmetrische und transitive Relation, \(\sim\)) as a tool to describe abstraction. One focuses on one or a few properties and leaves others out. All elements with a certain value of the property variable are equivalent and form an equivalence class. All equivalence classes make up the quotient space \(M/\sim\), which is a set of disjoint subsets of \(M\).

Abstract concepts can be combined again (Synthesis). This is the principle of creativity.