University:
John Jay College of Criminal Justice
Course:
MAT301 | Probability And Mathematical Statistics I
Academic year:

2021
Views:

318

Pages:

1
Author:

Aaqib Mehraj

RELATIONS AND FUNCTIONS1.0 IntroductionR-Superscript(1) = {(0, 0), (1, 1), (1, -1), (4, 2), (4, -2),(9, 3)}R is one among the 260 relations that could beWe are already familiar with the ideas ofdefined from A into B.'relations' and 'functions':Example 2Definition 1:A relation R, from a non - empty set A into aA is the set of all students of standard XII of ournon-empty set B is a subset of A x B.school and, B the set of all teachers of theschool. The relation, 'b is the MathematicsThus, R is a set of ordered pairs with firstteacher of a' for a E A and b E B can be specifiedelement in A and second element in B. A relation,by exhibiting all the ordered pairs (a, b) where ain the usual sense from a set A into a set BE A is a Mathematics student and b is a teacherdetermines a definite set of ordered pairs -doing Mathematics for a. What is the domain, theordered pairs with members of A and relatedrange and inverse of the relation?members of B as elements. In fact, a relationcould be described by enumerating such orderedSolutionpairs.A relation from A into B may be the empty subsetThe domain of the relation is the set of allof AxB or A x B itself.students of standard XII with Mathematics as oneof their subjects and the range is the set of allDefinition 2:teachers teaching Mathematics in Standard XII.The set of all first elements of R, a relation from AThe inverse relation from B to A is bR a iff a isinto B, is the domain of R and the set of allthe student of the Mathematics teacher b".second elements is the range of R.Definition 3:The domain is a subset of A and the range is aA function f from A into B is a relation on A intosubset of B. When (a, b) E R, we say 'a is R -B with domain A such that (a, b1) E f andrelated to b' and write a R b; when a is not(a,related to b, we write a R b. A relation from Ainto A is usually described as a relation on A.Letf be a function when (a, b) E f, we say b is theimage of a under f; and a is pre - image of bIf n (A) = p, n (B) = q, then the number ofunder f. We write b = f(a). A function may bepossible relations from A into B is 2P9specified by exhibiting the ordered pairsconstituting the function or by giving the rule for,The inverse of the relation R from A into B,fixing images of elements. In this further study ofdenoted by R-1 is the relation from B into A,the topics, we shall learn relations and functions,defined by R-1 = {(b, ):(a, b) E R}.with some special properties.Example 11. 1 RelationsLet A = {-2, -1, 0, 1, 2, 3},B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. We define aDefinition 4:relation R from A to B is R = {(-2, 4), (-1, 1),(0, 0), (1, 1), (2, 4), (3, 9)}. Write R in set builderEmpty relation from A into B is the empty subsetof A x B.form. What is the domain and range of R. WriteR-The domain, as well as the range of the relationSolutionis - none in A is related to any in B, under thisrelation.Clearly (a, b) E R iff b = a2eg: R = {(x, y): y = 0; y E N} is an emptyDomain of R = A, range of R={0,1,4,9}relation in N.

Relations and functions

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