R is the relation on Z (integers) given by xRy ( X is related to Y ) if 3 divides (x-y), what are the distinct elements of R?
Asked
Active
Viewed 99 times
-3
-
Can you come up with examples of elements of $R$? Non-examples? Is $1R5$? How about $3R9$? – Arthur Aug 26 '15 at 12:47
-
No, theres no given example. – user264567 Aug 26 '15 at 12:49
-
it has 4 distinct elements. – user264567 Aug 26 '15 at 13:01
-
Which four elements is that? – Arthur Aug 26 '15 at 13:04
-
possible duplicate of Two questions about equivalence relations – graydad Aug 26 '15 at 13:59
1 Answers
1
I would start by writing out what it means for 3 to divide a number, that is, $3 | n$ if and only if $\exists k$ s.t. $n = 3k$. When does $3 | (x - y)$ is the next question we should ask ourselves. As suggested in one of the comments on your post, often this is where it helps to come up with some examples, for instance, does $1R5$? does $5R8$ ? does $8R5$ ? Now that we've considered some examples, lets expand on the definition a little, when $3 | (x - y)$ we have that $\exists k$ s.t. $(x - y) = 3k$ or alternatively, x is related to y whenever their difference is a multiple of 3.
Benjamin Gadoua
- 373
-
-
Can you tell me two numbers who have a difference that is divisible by 3? – Benjamin Gadoua Aug 26 '15 at 13:30
-
-
-
the given is xRy therefore if 5R8 their difference is -3. is this still correct? – user264567 Aug 26 '15 at 15:14
-
Does 3 divide -3? If yes then 5R8 is an element of the relation. Check 10R4, clearly there are more than two elements. A plain description is the last statement of my answer. – Benjamin Gadoua Aug 26 '15 at 15:16
-