Question

Determine whether the relation O defined on ℤ is reflexive, symmetric, or transitive.
The relation O on ℤ is defined as follows: for all m, n ∈ ℤ, m O n ⇔ m – n is odd

Answer 1

Do you know what it means to be reflexive?

Answer 2

when there is (x,x) is an element in Z

Answer 3

whoops that was not proper english and didnt make sense but u got the jist of things

Answer 4

That's exactly what it means to be reflexive. So now, look at \(xOx\iff x-x\) is odd. Is x-x odd?

Answer 5

NOOOOOOOOOOOOO

Answer 6

since it will be 0 and 0 isnt odd

Answer 7

Perfect. So that means that it's not reflexive.

Answer 8

ummm well i think its syymetric

Answer 9

Onto symmetric. Suppose \(x-y\) is odd. Is \(y-x\) also odd?

Answer 10

yesssss

Answer 11

It sure is. So now we only need to look at transitive.

Answer 12

hmmmmm that one is a lil more difficult to figure out give me a sec

Answer 13

its not transitive

Answer 14

x=4 y=3 z=2

Answer 15

4-3=1
3-2=1
4-2=2

Answer 16

That's correct. And a counterexample to go along with it too.

Answer 17

In fact, you could prove that it's intransitive by saying that since \(x-y\) is odd, and \(y-z\) is odd, then \((x-y)+(y-z)=(x-z)\) must be even since odd+odd=even.

Answer 18

Rather, that would be antitransitivity, and not intransitivity.

Answer 19

huh? lol let me reread it

Answer 20

ohhhh i get it

Answer 21

YAY U R awesome KING GEORGE

Answer 22

Thank you :)

Answer 23

Good job figuring out most of this by yourself.