Question

A is the binary relation defined on real numbers as follows. For all real numbers 1, 39 XAy if and only if xy >0. Determine if A is reflexive, symmetric, transitive, antia symmetric.

Answer 1

- A in the binary relation defined on seal number as follows. Per all real numbers xy, xoy ich and only if xyzo. Determine oy A w reflexive, emmelic, transitive, anti- Я им е с We have x Ay ig and only in xy so Reflexie het XEIR be any real number Then x²30 Therefore xAX & XEIR real number 7 A in reflexive Symmetric het xy GIR be any areal number euch that xay We know xay it and only if x.480 cel and only if yx yo ach and only ij yax" . 3 3 Therefore xAy it and only if yAX a) A in symmetric

Transitive that het x,y,z GIR be any real numbers such XAY and YAZ XAY is ry yo and Yaz uj yzzo N Now XZ = (xyllyz). [-xys,o & yz 20 ya 707 PA ? o 2) xzzo Therefore xAzonor a r a) A in transitive Anti-symmetric By defination, A in anti eynmebic uy xay and yAx then to y Here let X= 1 and y=2 ton Then xy = 1.2=270 of Y = 21 22 do 2) XAY and YAX But xay a) A in not anti-symmetric