Logic is often described as the analysis of reasoning. The first systematic study of logic is due to the Greek philosopher Aristotle. In mathematical logic we study logic using mathematical tools. etc. etc.
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Abstract/Modern Algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field.
By abstracting away various amounts of detail, mathematicians have defined various algebraic structures that are used in many areas of mathematics. For instance,
almost all systems studied are sets, to which the theorems of set theory apply. Those sets that have a certain binary operation defined on them form magmas,
to which the concepts concerning magmas, as well those concerning sets, apply. We can add additional constraints on the algebraic structure, such as
associativity (to form semigroups); identity, and inverses (to form groups); and other more complex structures. With additional structure, more
theorems could be proved, but the generality is reduced. The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the
corresponding theories: for instance, the theorems of group theory may be used when studying rings (algebraic objects that have two binary operations
with certain axioms) since a ring is a group over one of its operations. In general there is a balance between the amount of generality and the richness
of the theory: more general structures have usually fewer nontrivial theorems and fewer applications.
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Physics is an empirical science, which means that it derives "fact" from careful observation through application of the scientific method. blah blah blah ...
Computer Science is a subfield of mathematics. It is often confused with IT, however, IT is the application of computer science to business processes, while computer science deals with abstract mathematics related to computer software and hardware.
Chemistry is the study of compounds, reactions etc. of compounds formed using the elements from the periodic table.
Biology is the study of living organisms.