Let n belong to the set of positive integers and let ~ be defined on the set of
ID: 2943178 • Letter: L
Question
Let n belong to the set of positive integers and let ~ be defined on the set of integers by r~s if and only if r-s is divisible by n, that is, if and only if, r-s=nq for some q in the set of integers. Here is question number 3: Determine whether the relation R defined on the set of real numbers R as follows: For all x,y in the set of real numbers R, xRy iff the absolute value of x-y <=3, is an equivalence relation. That is xRy iff [x-y]<=3. The brackets are a substitute for the absolute value bars. Nothing special about the brackets.Again, they are supposed to be absolute value bars.Explanation / Answer
GIVEN
X R Y ...DEFINED BY |X-Y|<=3
TPT
R IS AN EQUIVALENCE RELATION
1. REFLEXIVE..
IF A IS ANY ELEMENT IN THE SET THEN [A,A] IS AN ELEMENT IN THE RELATION ....
THAT IS TPT ..
|A-A|<= 3 ....OK.....
2. SYMMETRIC ....
IF A R B THEN TPT B R A ..
WE HAVE
|A-B| <= 3
BUT
|A-B|=|B-A| ..SO IT IS <= 3 ....OK
3.TRANSITIVE
IF A R B AND B RC TPT A R C
THAT IS WE HAVE
|A-B|<=3
|B-C|<=3
TO CHECK IF |A-C|<= 3 ...
IT IS NOT CORRECT SINCE TAKING A=4 , B = 2 AND C=0 WE HAVE
|A-B|=2<=0...OK
|B-C|=2<=0....OK
BUT
|A-C|=4>3 ...SO
THE GIVEN RELATION IS NOT REFLEXIVE ..
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