What is the meaning of 'x R x' and 'x R y --> x R z' in relation notation?

[cragar]

Homework Statement

Let R be a relation defined in a set A

If for all $x \in A$ we have x R x, we call R reflexive
what does it mean when they write x R x ?


And
if for all x,y,z in A we have (x R y and y R z) --> x R z, we call R transitive .
I am not sure what they mean by transitive

 

[LCKurtz]

Homework Statement

Let R be a relation defined in a set A

If for all $x \in A$ we have x R x, we call R reflexive
what does it mean when they write x R x ?


And
if for all x,y,z in A we have (x R y and y R z) --> x R z, we call R transitive .
I am not sure what they mean by transitive

Homework Statement

Let's look at a couple examples. Let A be the natural numbers. Let's define a relation R by saying mRn if m divides n. Let's see if it reflexive and transitive. To check reflexive we need:

For all n in A, nRn, which means n divides n. Obviously true, so this R is reflexive.

To check transitive we need if nRm and mRp then nRp, which in this case means:
If n divides m and m divides p, then n divides p. Can you see that is true so R is transitive.

Now look at a new R defined by mRn means m < n. Can you see this R is not reflexive but it is transitive? Does that help?

 

[cragar]

if m<n then n can't be less than m . so is that why it is not reflexive?
But it is transitive because m<n and there is another # such that m<n<z

 

[Dick]

if m<n then n can't be less than m . so is that why it is not reflexive?
But it is transitive because m<n and there is another # such that m<n<z

That m<n and n<m can't both be true proves it's not SYMMETRIC. It's not REFLEXIVE because n<n isn't true. Do you see how that's related to the x R x?

 

[cragar]

I think i see now, how do we pronounce x R x
and also is this set transitive {(1,2), (2,3) , (1,3 )}

 

[Dick]

I think i see now, how do we pronounce x R x

"x is related by R to x". If R is "less than" than "x is less than x". If R is "divisible" then "x is divisible by x".

 

[cragar]

ok thanks for your response .
I was just wondering if this set I made up was transitive. to check my understanding.
and also is this set transitive {(1,2), (2,3) , (1,3 )}

 

[Dick]

ok thanks for your response .
I was just wondering if this set I made up was transitive. to check my understanding.
and also is this set transitive {(1,2), (2,3) , (1,3 )}

Yes, it is.

 

[cragar]

ok i think i understand now,