Consider the \"punctured plane,\" that is RxR--{(0,0)}. Define an equivalence re
ID: 2941476 • Letter: C
Question
Consider the "punctured plane," that is RxR--{(0,0)}. Define an equivalence relationR by (x1, y1)R(x2, y2) when there is some positive real number a such that ax1 = ax2
and ay1 = y2. In other words: (x, y)R(ax,ay) for all positive reals a.
(a) Find all points (x, y) such that (0,1)R(x,y). Find all points (x,y) such that
(1,1)R(x,y).
(b) Show that R is an equivalence relation.
(c) Briefly state what the equivalence classes are.
(d) Find a system of representatives: a set with exactly one element from each
equivalence class.
Explanation / Answer
Consider the "punctured plane," that is RxR--{(0,0)}. Define an equivalence relation
R by (x1, y1)R(x2, y2) when there is some positive real number a such that ax1 = ax2
and ay1 = y2. In other words: (x, y)R(ax,ay) for all positive reals a.
(a) Find all points (x, y) such that (0,1)R(x,y). Find all points (x,y) such that
(1,1)R(x,y).
FROM THE GIVEN DEFINITION...
[0,1]R[X,Y]...THEN X=0*A=0.....AND Y=1*A=A...
THAT IS
[0,1]R[0,A] WHERE A IS ANY REAL NUMBER
SIMILARLY
[1,1]R[A,A] WHERE A IS ANY REAL NUMBER.
(b) Show that R is an equivalence relation.
1. REFLEXIVE
FOR ANY ELEMENT [X,Y] WE HAVE
[X,Y]R[AX,AY]...WHERE A IS ANY REAL NUMBER TAKING A=1 WE GET
[X,Y]R[X,Y]....SO REFLEXIVE
2.SYMMETRIC
WE HAVE IF [X,Y]R[X',Y']...
X'=AX.......Y'=AY...
AND SO ....[X',Y']R[X,Y].....SINCE X=BX' AND Y=BY'
WHERE.....B=1/A...[ASSUMING A IS NOT EQUAL ZERO]
IF A=0 ...THEN [X,Y]R[0,0]...THEN WE CAN NOT SAY [0,0]R[X,Y]
SO THIS HAS TO BE EXCLUDED IN THE DEFINITION TO MAKE IT EQUIVALENCE RELATION.
3.TRANSITIVE
WE HAVE IF [X,Y]R[X',Y']...[X',Y']R[X'',Y'']...THEN
X'=AX.......Y'=AY..........X''=BX'.......Y''=BY'
AND SO ....[X'',Y'']R[X,Y].....SINCE X=CX'' AND Y=CY''
WHERE.....C=1/(AB)...[ASSUMING A AND B ARE NOT EQUAL ZERO]
HENCE IT IS AN EQUIVALENCE RELATION IF IT IS CLARIFIED THAT A IS NOT EQUAL TO ZERO IN THE DEFINITION......In other words: (x, y)R(ax,ay) for all positive reals a....
FOR A NOT EQUAL TO ZERO..
(c) Briefly state what the equivalence classes are.
[X,Y]R[2X,2Y]......THEY REPRESENT SAME CURVES WHEN PLOTTED AS GRAPH OF Y VS X
(d) Find a system of representatives: a set with exactly one element from each
equivalence class.
YOU MEAN EXAMPLES OF ABOVE ???
[X,PX+Q] R [AX,APX+AQ].....[X,2X+3]R[2X,4X+6] ARE ST.LINES
[X,X*X]R[AX,AX*X]...ETC...