Equivalence Relations: Let S={a,b,c,d,e} and R= {(a,a),(a,c),(a,e),(b,b),(b,d),...}

[HugeLag]

Let S={a,b,c,d,e} and R= {(a,a),(a,c),(a,e),(b,b),(b,d),( c,a), (c,c),(c,e),(d,b),(d,d),(e,a),(e ,c),(e,e)}

Is R is reflexive, symmetric, transitive and an equivalence relation ?

TRUE/FALSE?

My Method/Knowledge/Answer: -

Knowledge-

Ok so I know the three classes and their rules are: -

Reflexive = a~b
Symmetric = a~b and b~a
Transitive = If a~b and b~c then a~c.

Method-

For example with

R={(a,a)} = This would be Reflexive (Because a is equal to a)
R={(a,c)}/{(c,a)} = This would be Symmetric (Because a is equal to b and b is equal to a)
R={(a,e)}/{(e,a)} = This would be Transitive (Because if a is equal to b and b is equal to c then a is equal to c)

R={(c,c)} = This would be Reflexive (Because a is equal to a)
R={(c,a)}/{(a,c)} = This would be Symmetric (Because a is equal to b and b is equal to a)
R={(c,e)}/{(e,c)} = This would be Transitive (Because if a is equal to b and b is equal to c then a is equal to c)

R={(e,e)} = This would be Reflexive (Because a is equal to a)
R={(e,a)} = This would be Symmetric (Because a is equal to b and b is equal to a)
R={(e,c)}/{(c,e)} = This would be Transitive (Because if a is equal to b and b is equal to c then a is equal to c)

Now this is why I think the whole answer is false is this next part

R={(b,b)} = This would be Reflexive (Because a is equal to a)
R={(b,d)}/{(d,b)} = This would be Symmetric (Because a is equal to b and b is equal to a)
For transitive their would be nothing? Am I right? (Just because their is no transitive here would it make the whole answer false?)



FYI: Sorry for the lengthy post just want to put my thoughts to paper, so people dont think I am here for quick answers.

 

[pka]

Let S={a,b,c,d,e} and R= {(a,a),(a,c),(a,e),(b,b),(b,d),( c,a), (c,c),(c,e),(d,b),(d,d),(e,a),(e ,c),(e,e)}
Is R is reflexive, symmetric, transitive and an equivalence relation ?
TRUE/FALSE?

From this posting, it is hard to see that you know much about relations on a set.
Do you know what the diagonal is? \(\displaystyle \Delta_S=\{(x,x) : x \in S\}=\{(a,a),~(b,b),~(c,c),~(d,d),~(e,e)\}\)
If \(\displaystyle \Delta_S\subset \mathcal{R}\) then the relation is reflexive. In other words, is each term of \(\displaystyle S\) related to itself.

The relation is symmetric if \(\displaystyle \mathcal{R}=\mathcal{R}^{-1}\). Check to see if for each pair in \(\displaystyle \mathcal{R}\) its inverse is also in \(\displaystyle \mathcal{R}\).

Transitive is always the hard one. But you seem to understand it a bit.
If \(\displaystyle (a,b)\in\mathcal{R}~\&~(b,c)\in\mathcal{R}\) then it must be that \(\displaystyle (a,c)\in\mathcal{R}\)

 

[HallsofIvy]

Ok so I know the three classes and their rules are: -

Reflexive = a~b
Symmetric = a~b and b~a

Neither of these is correct.
"Reflexive" means that "a~ a for all a in the set"
"Symmetric" means that "if a~b then b~ a"

Transitive = If a~b and b~c then a~c.

Okay, this is correct.

 

[Steven G]

Let S={a,b,c,d,e} and R= {(a,a),(a,c),(a,e),(b,b),(b,d),( c,a), (c,c),(c,e),(d,b),(d,d),(e,a),(e ,c),(e,e)}

Is R is reflexive, symmetric, transitive and an equivalence relation ?

TRUE/FALSE?

My Method/Knowledge/Answer: -

Knowledge-

Ok so I know the three classes and their rules are: -

Reflexive = a~b
Symmetric = a~b and b~a
Transitive = If a~b and b~c then a~c.

Method-

For example with

R={(a,a)} = This would be Reflexive (Because a is equal to a)
R={(a,c)}/{(c,a)} = This would be Symmetric (Because a is equal to b and b is equal to a)
R={(a,e)}/{(e,a)} = This would be Transitive (Because if a is equal to b and b is equal to c then a is equal to c)

R={(c,c)} = This would be Reflexive (Because a is equal to a)
R={(c,a)}/{(a,c)} = This would be Symmetric (Because a is equal to b and b is equal to a)
R={(c,e)}/{(e,c)} = This would be Transitive (Because if a is equal to b and b is equal to c then a is equal to c)

R={(e,e)} = This would be Reflexive (Because a is equal to a)
R={(e,a)} = This would be Symmetric (Because a is equal to b and b is equal to a)
R={(e,c)}/{(c,e)} = This would be Transitive (Because if a is equal to b and b is equal to c then a is equal to c)

Now this is why I think the whole answer is false is this next part

R={(b,b)} = This would be Reflexive (Because a is equal to a)
R={(b,d)}/{(d,b)} = This would be Symmetric (Because a is equal to b and b is equal to a)
For transitive their would be nothing? Am I right? (Just because their is no transitive here would it make the whole answer false?)



FYI: Sorry for the lengthy post just want to put my thoughts to paper, so people dont think I am here for quick answers.

You really need to state things correctly-even *if* you understand them.

1st you stated that R= {(a,a),(a,c),(a,e),(b,b),(b,d),( c,a), (c,c),(c,e),(d,b),(d,d),(e,a),(e ,c),(e,e)} .
Then you stated that R={(a,a)} , R={(a,c)}/{(c,a)}, R={(a,e)}/{(e,a)}, R={(c,c)} , R={(c,a)}/{(a,c)}, R={(c,e)}/{(e,c)}, R={(e,e)}, R={(e,a)} and R={(e,c)}/{(c,e)}

I just have one question about the above. What does R equal???!!!!

Then you wrote R={(c,c)} = This would be Reflexive (Because a is equal to a). I guess that you meant to say that (c,c) is reflexive because a is RELATED (not equal to) a. Again, c is related to c BECAUSE a is related to a? I do not think so. c being related to c has nothing to do with whether or not a is related to a. c is related to c BECAUSE c is related to c. That is (c,c) is in R.


It has already been stated why R is reflexive. It is because {(a,a), (b,b), (c,c), (d,d), (e,e)} is in R

Just write real math.

 

[HallsofIvy]

I have decided to come back and go through the whole post because you clearly do NOT "understand"!

Let S={a,b,c,d,e} and R= {(a,a),(a,c),(a,e),(b,b),(b,d),( c,a), (c,c),(c,e),(d,b),(d,d),(e,a),(e ,c),(e,e)}

Is R is reflexive, symmetric, transitive and an equivalence relation ?

TRUE/FALSE?

My Method/Knowledge/Answer: -

Knowledge-

Ok so I know the three classes and their rules are: -

Reflexive = a~b
Symmetric = a~b and b~a

No, as I said before, a relation is "reflexive" if and only if "a~ a for all a in S" and a relation is symmetric if and only if "if a~b then b~a".

Transitive = If a~b and b~c then a~c.

Method-

For example with

R={(a,a)} = This would be Reflexive (Because a is equal to a)

No, it would NOT be reflexive because we do NOT have b~b, c~c, d~d, and e~e.
It is both symmetric and transitive. Do you understand that you should answer "true" or "false" for all three? And it is not an equivalence relation because not all three are true.

R={(a,c)}/{(c,a)} = This would be Symmetric (Because a is equal to b and b is equal to a)

I don't know what that "/" is intended to mean. If you mean simply {(a, c), (c, a)} then, yes, it is symmertric. Although you reason, in parentheses, makes no sense because there is NO "b" in the set. Further you should not say "equal" here. "Equal" is an equivalence relation but not all relations or even equivalence relations are "equals". Also this it neither "reflexive" nor "transitive". (Because it contains both (a, c) and (c, a) it would have to contain (a, a) and (c, c) to be transitive.)

R={(a,e)}/{(e,a)} = This would be Transitive (Because if a is equal to b and b is equal to c then a is equal to c)

This is exactly the same as the previous example! There is no "b" in this so there is no reason to mention "b". This is symmetric but neither reflexive nor transitive. If it were transitive then, because it contains (a, e) and (e, a), it would have to contain (a, a) and (e, e).

R={(c,c)} = This would be Reflexive (Because a is equal to a)

No, it is NOT reflexive because it does NOT contain (a, a), (b, b), (d, d), and (e, e). In order to be reflexive a relation must contain ALL of (a, a), (b, b), (c, c), (d, d), and (e, e).

R={(c,a)}/{(a,c)} = This would be Symmetric (Because a is equal to b and b is equal to a)
R={(c,e)}/{(e,c)} = This would be Transitive (Because if a is equal to b and b is equal to c then a is equal to c)

These two are exactly the same as the other two that had that "\" in them! It is symmetric but neither reflexive nor transitive.

R={(e,e)} = This would be Reflexive (Because a is equal to a)

No, it is NOT reflexive because we do NOT have (a, a), (b, b), (c, c), or (d, d).

R={(e,a)} = This would be Symmetric (Because a is equal to b and b is equal to a)

No, it is NOT symmetric because it has (e, a) but NOT (a, e). It is not reflexive but it is (trivially) transitive.

R={(e,c)}/{(c,e)} = This would be Transitive (Because if a is equal to b and b is equal to c then a is equal to c)

Again, I assume that you mean R= {(e,c), (c, e)}. Just like the four above, it is symmetric but not reflexive or transitive.

Now this is why I think the whole answer is false is this next part

The "whole answer" to what question?

R={(b,b)} = This would be Reflexive (Because a is equal to a)

Again this is exactly like the very first exercise, R= {(a, a)}. It is symmetric and transitive but it is NOT reflexive.

R={(b,d)}/{(d,b)} = This would be Symmetric (Because a is equal to b and b is equal to a)

Yet again, assuming that you mean R= {(b, d), (d, b)} this is symmetric but not reflexive nor transitive.

For transitive their would be nothing? Am I right? (Just because their is no transitive here would it make the whole answer false?)

"Whole answer" to what question?

FYI: Sorry for the lengthy post just want to put my thoughts to paper, so people dont think I am here for quick answers.

[/quote]
Thank you for that.