Set



Introduction to Set



Subjects to be Learned

Contents

The concept of set is fundamental to mathematics and computer science. Everything mathematical starts with sets. As we see later, relationships between two objects are represented as a set of ordered pairs of objects, the concept of ordered pair is defined using sets, natural numbers, which are the basis of other numbers, are also defined using sets, the concept of function, being a special type of relation, is based on sets, and graphs and digraphs consisting of lines and points are described as an ordered pair of sets.

Though the concept of set is fundamental to mathematics, it is not going to be defined rigorously here. Instead we rely on everyone's notion of "set" as a collection of objects or a container of objects. In that sense "set" is an undefined concept here. Similarly we say an object "belongs to " or "is a member of" a set without rigorously defining what it means. This approach to set theory is called "naive set theory" as opposed to "axiomatic set theory". The naive set theory produces paradoxes such as Russell's paradox, hence it is not consistent, meaning that a statement which should be true may not be proven true following the naive set theory. However, it is simpler and practically all the results we need can be derived within the naive set theory. Thus we shall be following this naive set theory in this course. "An object(element) x belongs to a set A"  is symbolically represented by  "x A" .


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