Partial preview of the text
Download Mathematics Preliminiries (sets, subsets, relations, binary composition, prime numbers and more Study notes Mathematics in PDF only on Docsity!
Mathematics "Preliminaries" (sets, subsets, equivalence classes, permutations, cyclic permutations, disjoint permutations, congruences, relations, binary composition, prime numbers, etc) detailed notes for 12 Class, BS Mathematics, MSc Mathematics, and competitive exams Preliminaries Introduction In this chapter we remind or acquaint the reader about some basic concepts in mathematics that we reckon the reader would already be in the know of, but in case not, we strongly recommend one to glance through the contents of this chapter before venturing into the subsequent text. We basically explain the concepts of sets along with operations in sets and then go on to define the all-important notion of a mapping/function(and permutations), which finally lead us to the definition of a binary composition/operation. In the second half of this chapter we take a peek at the results from number theory and try to discuss most of the relevant results that could be useful in the main text. Having done this chapter, one is fully equipped to understand and grasp the subsequent material that follows. Sets The notion of a set is most fundamental in Mathematics, but it is not our endeavour in this text to enter into the axiomatic study of set theory. We'll, instead, borrow the word ‘set’ from the language and be content to refer to it as a collection of objects. To give it a more precise shape, by a set, we will mean a collection of objects such that given any object, it is possible to ascertain whether that object belongs to the given collection or not. For instance, we can talk of set of all natural numbers, set of all students in a particular class, etc. If x is an element (member) of a set A we say x belongs to A and express it as x € A. If y is not a member of A we say y does not belong to A and write y ¢ A. We shall use capital letters A, B, X, Y etc. for denoting sets and small letters, a, b, c, x, y etc. for the elements (or members or objects). Two sets A and B are said to be equal if they contain precisely the same elements and we write A = B. A set can be described in various ways. For example, if A is the set containing 1, 2, 3, 4, 5, 6, we can write it as A = {1, 2, 3, 4, 5, 6} A = £1, 2, ...... , 6} A={xeEeN|xs6} where N is set of all natural numbers. The last notation reading as: those x in the set of natural numbers which satisfy the property that x < 6. 
