0

Let $\mathbf{X}$ be the set of all residents in New Jersey. Determine which are equivalance relations:

a) $x\sim y$ provided $y$ has the same natural parents as $x$

b) $x\sim y$ provided $y$ lives within 5 miles of $x$

c) $x\sim y$ provided $y$ has same birth date as $x$

d) $x\sim y$ provided $y$ is the brother of $x$

I know i have to either prove all reflective, symmetric and transitive properties or disprove one. I'm struggling how to show my reasoning for it.

For example for (b) I can see it fails the transitive prop as say $z$ could be 5 miles east of $y$ which is 5 miles east of $x$ thus $x$ isn't within 5 miles of $x$ and (d) is not symmetric as $x$ could be the sister of $y$.

Thanks

Winther
  • 24,478
user307533
  • 37
  • 3
  • What's you specific problem? If you are uncertain about C - this is an equivalence relation because all three rules are satisfied - reflexivity, symmetry and transitivity, perhaps you could imagine a building with 366 different rooms, and you assign people to a particular room based up on their birthday, so at the end of the day, each room would only contain people with the same birthday. Similarly, if you met somebody, you would know exactly which room to send them too. Does that help? – user247608 Jan 25 '16 at 01:25
  • im not sure on how to prove they are a equivalence relation for a and b – user307533 Jan 25 '16 at 01:26
  • 1
    B is not an equivalence relation, if x and y live within 5 miles of each other, and y and z live within 5 miles, its not true that x and z live within 5 miles, they may be up to 10 miles apart, so the transivity doesn't hold. – user247608 Jan 25 '16 at 01:35
  • A is an equivalence relation - reflexivity might be a bit of word game - if x's natural parents are A and B, then x~x is true because the left side has the same natural parents as the right side. Similarly if x ~ y, then also y ~ x, and finally the transitivity holds, if x ~ y and y ~ z then x and z must have the same natural parents too. – user247608 Jan 25 '16 at 01:38
  • I didn't mean to say B , but anyway i understand it now thanks for your help – user307533 Jan 25 '16 at 01:39

0 Answers0