Hi /sci/, it's 2am and I have a discrete maths exam tomorrow

>>8575685
>Can a binary relation be reflexive,symmetric,anti symmetric AND transitive at the same time ?

yes, but symmetric and antisymmetric at the same time makes it boring, this works with the usual = on whatever set of numbers you want (reals, rationals, integers...)

>Plus a simple explanation of transitive relations would be greatly appreciated
ordering relations work here (like < or >) since a< b and b<c imply a<c

>if aRb and bRc then aRc but are there any special cases where a relation is transitive without this being the case
what do you mean here, this is what transitive means

>>8575685
Yes, a relation full of elements like (x,x)

It is reflexive
It is symmetric trivially
It is anti symmetric
It is transitive trivially

What is a transitive relation? Well, a relation is transitive if and only if:
IF (a,b) is an element of the relation and (b,c) is an element of the relation then (a,c) must be an element of a relation.

Why is the relation I constructed trivially transititive? Because no such scenario even exists. There are just elements like (1,1) and (2,2) so the only statement of transitivity in the relation is:

(1,1) is in the relation and (1,1) is in the relation so (1,1) is in the relation... and it is. Trivially

>>8575697
Thank you anon
so a reflexive relation can be considered transitive ? If there were nothing but loops ?

>>8575697
ordering relations work here

TERRIBLE explanation. Orders are defined as relations that are transitive, anti symmetric and either reflexive or non-reflexive depending on the type of order.

>>8575704
>so a reflexive relation can be considered transitive ?
yes, tons of them, when they're also symmetric they're called equivalence relations
https://en.wikipedia.org/wiki/Equivalence_relation

>>8575700
Thank you anon , much appreciated

>>8575700
Then when is a relation not transitive ?

>>8575720
https://en.wikipedia.org/wiki/Transitive_relation
>"is the mother of" on the set of all people is not transitive since your mother's mother isn't your mother

>>8575720
When it contradicts that definition
Let me give you an example:

{ (a,b) , (b,c) }

that is not transitive. Also

{ (a,b) , (b,c) , (a,c) , (b,d) }

Exercise: Why is this second relation not transitive?

>>8575728
Exactly,why?

>>8575728
Because there's no c,d ?

>>8575732
Ok, at this point it serves thinking like a computer. Remember the definition of transitivity is universal.

FOR ALL pairs, transitvity must occur for a set to be called transitive. Even if there is transitivity all over the place except for 1 case then you function is non-transitive.

So lets go element by element and see

(a,b). is there. (b,c) is there. Therefore, by the definition of transitivity, there must be some (a,c) out there.

Oh, look it is there. Good to know.

Next element

(b,c) is there. (a,c) is there. Nothing happens here

(b.c) is there, (b.d) is there. Nothing happens there

(a,b) is there. (b,d) is there. TRANSITIVE ALARM WOOOH WOOH. If these two elements are in the relation and we say the relation is transitive then there must be an (a,d) element somewhere out there.

Oh, there isn't.

Then it is not transitive.

QED
E
D

>>8575733

No, because there is no (a,d). The fact that you got this wrong says you better go back to the logic of universal operators and implications and then go burn the definition of transitivity in your retina.

>>8575740
>TRANSITIVE ALARM WOOOH WOOH.
cringe

>>8575743
ay fuq u.

>>8575740
Thank you so much anon
sorry for all the trouble