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CBSE - XII Science - Mathematics - Relations and Functions

EXPERTS I SHOULDN'T POST 3 QUERIES AT A TIME BUT ITS URGENT I HAVE TEST TOMORROW. SOLVE ALL 3 QUERIES.

1.Let Q+ denote the set of all positive rational numbers.

A relation R on Q+ × Q+ is defined as .  Show that R is an equivalence relation.


2.  Let A = R − {0} and B = × R. .Let the binary operation * on B be defined as    (a, b) * (c, d)    = http://cbse.meritnation.com/img/paper/1/12/1/807/chapter%201_html_md5f7c8d.gif.(i) Find the identity element of B with respect to the   operation *. (ii) Show that set B is invertible with respect to the operation *.Also, find the     inverse of (−5, 3).


3.A function f: Z → A, wherehttp://cbse.meritnation.com/img/paper/1/12/1/807/chapter%201_html_7f0eac49.gifis defined ashttp://cbse.meritnation.com/img/paper/1/12/1/807/chapter%201_html_m35c6f42.gif.

Without finding the actual inverse of f, show that f is  bijective.


Asked by nikitaayyagari 28th April 2016, 1:29 PM
Answered by Expert

Answer:

begin mathsize 16px style 1.
G i v e n space t h a t space R space i s space a space r e l a t i o n comma space R colon Q to the power of plus rightwards arrow Q to the power of plus space s u c h space t h a t comma space open parentheses a comma b close parentheses R open parentheses c comma d close parentheses left right double arrow open parentheses 1 over a plus 1 over d close parentheses equals open parentheses 1 over b plus 1 over c close parentheses
W e space n e e d space t o space p r o v e space t h a t space R space i s space a n space e q u i v a l e n c e space r e l a t i o n.
R e f l e x i v e colon
W e space h a v e comma
1 over a plus 1 over b equals 1 over a plus 1 over b
T h u s comma space open parentheses a comma b close parentheses R open parentheses a comma b close parentheses.
T h e r e f o r e comma space R space i s space r e f l e x i v e.

S y m m e t r i c colon
C o n s i d e r space open parentheses a comma b close parentheses R open parentheses c comma d close parentheses
rightwards double arrow open parentheses 1 over a plus 1 over d close parentheses equals open parentheses 1 over b plus 1 over c close parentheses
rightwards double arrow open parentheses 1 over b plus 1 over c close parentheses equals open parentheses 1 over a plus 1 over d close parentheses
rightwards double arrow open parentheses 1 over c plus 1 over b close parentheses equals open parentheses 1 over d plus 1 over a close parentheses
rightwards double arrow open parentheses c comma d close parentheses R open parentheses a comma b close parentheses
S i n c e space open parentheses a comma b close parentheses R open parentheses c comma d close parentheses rightwards double arrow open parentheses c comma d close parentheses R open parentheses a comma b close parentheses comma space R space i s space s y m m e t r i c.

T r a n s i t i v i t y colon
C o n s i d e r space open parentheses a comma b close parentheses R open parentheses c comma d close parentheses comma space a n d space open parentheses c comma d close parentheses R open parentheses e comma f close parentheses
rightwards double arrow open parentheses 1 over a plus 1 over d close parentheses equals open parentheses 1 over b plus 1 over c close parentheses space a n d space open parentheses 1 over c plus 1 over f close parentheses equals open parentheses 1 over d plus 1 over e close parentheses... left parenthesis 1 right parenthesis
rightwards double arrow open parentheses 1 over a plus 1 over d close parentheses equals open parentheses 1 over b plus 1 over c close parentheses space a n d space 1 over f equals open parentheses 1 over d plus 1 over e close parentheses minus 1 over c.. left parenthesis 2 right parenthesis

N o w space c o n s i d e r comma
1 over a plus 1 over f equals 1 over a plus open parentheses 1 over d plus 1 over e close parentheses minus 1 over c space space space space space open square brackets because s u b s t i t u t i n g space f r o m space e q u a t i o n space left parenthesis 2 right parenthesis close square brackets
rightwards double arrow 1 over a plus 1 over f equals open parentheses 1 over a plus 1 over d close parentheses plus 1 over e minus 1 over c space space space
rightwards double arrow 1 over a plus 1 over f equals open parentheses 1 over b plus 1 over c close parentheses plus 1 over e minus 1 over c space space space space space space open square brackets because s u b s t i t u t i n g space f r o m space e q u a t i o n space left parenthesis 1 right parenthesis close square brackets
rightwards double arrow 1 over a plus 1 over f equals 1 over b plus 1 over c plus 1 over e minus 1 over c
rightwards double arrow 1 over a plus 1 over f equals 1 over b plus 1 over e
rightwards double arrow open parentheses a comma b close parentheses R open parentheses e comma f close parentheses
T h u s comma space open parentheses a comma b close parentheses R open parentheses c comma d close parentheses comma space a n d space open parentheses c comma d close parentheses R open parentheses e comma f close parentheses space rightwards double arrow open parentheses a comma b close parentheses R open parentheses e comma f close parentheses
T h e r e f o r e comma space R space i s space t r a n s i t i v e.
T h e r f o r e comma space R space i s space a n space E q u i v a l e n c e space R e l a t i o n.





end style
Answered by Expert 29th April 2016, 9:01 AM

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