Certain fundamental mathematical notions play an important role in many parts of mathematics:

• Here you can find information on the concepts countable and uncountable, which have to do with the size of sets.
• The well-known associative, commutative, and distributive laws are given an extensive treatment here.
• A mathematical statement that requires no proof, or cannot be proved, is an axiom. Axioms are the foundation stones of mathematics.
• In mathematics, different types of problems can be tackled with different kinds of proofs. Choosing the right method of proof can simplify the approach to a problem.
• Here you can obtain information on the concepts of vector and dimension. 
• In considering the size of a set, it is of particular importance whether it is finite or infinite. These two categories of sets have fundamentally different properties.  
• Information about what is meant by existence in mathematics can be found here.  
• Problems involving finding maximum and minimum values are called extreme value problems.
• When one is interested in the “arrangement” of elements in a set, one speaks of order in a set.
• Also the notion equality is of particular importance (see also the article of E. Behrends from "Five-Minute Mathematics").
• The proof method known as induction makes it possible to prove statements that apply to all the natural numbers simultaneously. (See also the article of E. Behrends from "Five-Minute Mathematics").