Question

There are as many equivalence classes as there are which of the following? (Select all that apply.)

Answer Choices:

 

A. distinct horizontal lines in the plane

 

B. distinct integers

 

C. distinct real numbers

 

D. distinct vertical lines in the plane

 

E. distinct lines in the plane whose coordinates equal each other

Answer 1

Equivalence Relation: must be reflexive, symmetric, transitive

   A)  x ~ y if x and y are horizontal lines in the same plane

    x~x and y~y clearly hold, so reflexive

   x~y means that horizontal line x and y are in the same plane.  Then y and x are in the same plane, so symmetric.

x~y and y~z means that horizontal lines x, y, and z are all  in the same plane. then x and z are in the same plane.    therefore x~z, so transitive

B) x~y if x and y are integers

 x~x and y~y clearly hold, so reflexive,    because x=x and y=y are integers

if x~y, then x and y are integers.  So then y and x are integers. Then y~x, so symmetric.

if x~y and y~z, then x,y,z are all integers. Then

SPECIFICALLY, x and z are integers, so x~z, which proves the transitive

 

 C) same as B except they are REAL NUMEBRS instead of integers.

 

 D) same as A except they are VERTICAL LINES instead of horizontal lines. Again the logic is the same

E) Lines having the same co-ordinates means it is the same line. So there is only one element in this equivalence relation. I would not select this one.

so equivalence relation have..

A. distinct horizontal lines in the plane

B. distinct integers

C. distinct real numbers

D. distinct vertical lines in the plane