Relation on X: Symmetry, Reflexivity & Transitivity

[sam0617]

Let X = { a, b, c }

X x X = { (a,a), (b,b), (c,c) }
{ (a,b), (b,a), (a,c), (c,a) }
{ (b,c), (c,b) }

1. Symmetric but not reflexive or transitive:
R = { (a,b), (b,a), (a,a), (b,c), (c,b) }
How come this is right? Isn't aRb, bRa imply aRa? isn't that transitive? is it because (b,c,), (c,b) is there but not (b,b) the reason why R is not transitive?

I ask because the 2nd question is confusing. Here it is:
2. Symmetric and transitive but not reflexive:
R= { (a,a), (a,b), (b,a), (b,b) }
See how aRb, bRa implies aRa so therefore it's transitive? How come it doesn't hold for the 1st question??

Thank you for any help.

 

[micromass]

Transitive means that for ALL x,y,z:

$$xRy~\text{and}~yRz~\Rightarrow~xRz$$

It must holds for ALL.

For the first one, it isn't transitive since if you take x=z=a and y=x, then you see that the above is not satisfied. So it doesn't hold for ALL x,y,z. It does hold for some x,y,z. But it does hold for some. But some isn't enough to imply transitivity.

In (2), it does hold for ALL, so it is transitive.

 

[sam0617]

Transitive means that for ALL x,y,z:

$$xRy~\text{and}~yRz~\Rightarrow~xRz$$

It must holds for ALL.

For the first one, it isn't transitive since if you take x=z=a and y=x, then you see that the above is not satisfied. So it doesn't hold for ALL x,y,z. It does hold for some x,y,z. But it does hold for some. But some isn't enough to imply transitivity.

In (2), it does hold for ALL, so it is transitive.

I'm sorry, I don't understand what the x=z=a and y=x then it wouldn't satisfy above.
Could you explain more?

EDIT: Then to make question 1 transitive, all I would have to add is (b,b) ?

 

[micromass]

Sorry, typo. I meant that if x=z=b and y=c, then it isn't true that bRc and cRb and bRb.

Adding (b,b) would not make it transtive.

Indeed, we also don't have

aRb and bRc => aRc

since (a,c) is not in the relation.

 

[blue_raver22]

I can explain the concepts of symmetry, reflexivity, and transitivity in relation to the given content on X.

Symmetry refers to a relation where if (a,b) is in the relation, then (b,a) is also in the relation. This means that the order of the elements does not matter. In the given example, both R in the first and second question are symmetric because for every pair (a,b) in R, the pair (b,a) is also present.

Reflexivity refers to a relation where every element is related to itself. In the first question, R is not reflexive because (a,a) and (b,b) are not present in R. However, in the second question, R is reflexive because (a,a) and (b,b) are present in R.

Transitivity refers to a relation where if (a,b) and (b,c) are in the relation, then (a,c) is also in the relation. This means that if there is a chain of related elements, then the first and last elements are also related. In the first question, R is not transitive because although (a,b) and (b,c) are present, (a,c) is not. This is because (b,c) and (c,b) are present, but (b,b) is not, which breaks the transitive property. In the second question, R is transitive because for any elements a, b, and c, if (a,b) and (b,c) are present, then (a,c) is also present.

In summary, the first question is an example of a symmetric relation that is not reflexive or transitive, while the second question is an example of a symmetric and transitive relation that is also reflexive. The difference lies in the specific pairs of elements that are included in the relation.