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Ordering on Natural Numbers Previously, we have defined natural numbers that was a side effect of our spurious definition, e.g. . Also, we have the following simple definition of order on natural numbers:

Arithmetic We can apply recursion theorem, from the previous section, to define addition and multiplication on . Suppose we want a function such that is the result of adding 5 to n. Then must satisfy the

Consider the situation where I want you to guess the function such that I only give you two information; that is, a starting value a function such that for all This gives away all the information where

Hi! I'm Giuseppe Peano, I introduce Peano system to make the life of math students miserable. Source In 1889, Peano published a study giving an axiomatic approach to the natural numbers, showing how properties

In mathematics, there are two ways of introducing new objects for study, axiomatic approach (e.g. the one we have used for sets) The concept of set is one of our primitive notions, and we have adopted

Elements of Set Theory: Ordering Relations From the first section of this chapter, we've covered about the ordering relation of the ordered set Now we want to consider the ordering relation on other sets.

Consider a set A given by a figure in (a), say we want to partition it into six boxes as in (b). For example, take, we can partition into six parts: By partition we mean, dividing the box into a similar

We've encountered in previous section Cartesians products of two sets, this time we will show that we can form something like the Cartesian product but of infinitely many sets, provided that the sets are

In most calculus books a function is introduced as a rule that assigns to each object in a certain set (domain) a unique object in a possibly different set (range). Typical example: The action of this

Elements of Set Theory: n-ARY relations You might be wondering why do relations only have to do with ordered pairs. We can actually extend the ideas of ordered pairs to the case of ordered triples, and,

Elements of Set Theory: Relations First let's consider some examples. The ordering relation < on the set is one example. We say that < relates each number to each of the larger numbers. We can visualize

Elements of Set Theory: Ordered Pairs Consider the following pair set: this can be thought as an unordered pair. Consider another pair set with additional information: , where 1 is the first component,

Elements of Set Theory: Algebra of Sets The two basic operation on sets are operation of union operation of intersection We also have the operation relative complement of B in A: (some books denotes relative

Specifying attributes and Behaviors We now have now added object-oriented terminology in our arsenal of definitions. We defined objects as instances of classes, where it is an object of its own set of

Previously, we have the union axiom in its preliminary form, this union operation allowed us to form the union of two sets. And by repeating the process, we can form the union of three sets or the union

Now, we are done with the introduction chapter of the book, Elements of Set Theory. Again, I would like to emphasize the importance of the axiomatic method which we will employ throughout the book. And

The concept of a set is very basic and natural, and has been used in mathematical writings since ancient times. But the theory of abstract sets, as objects to be studied for their own interest, was originated

Elements of Set Theory: Notation To denote sets, we will use a variety of letters, both lowercase (a,b,...), uppercase (A,B,...), and even script letters and Greek letters. Letters can be embellished with