problem about reflexive relation

Hi

I have this problem in the topic of relations.

Suppose is a relation on , and define a relation

on as follows.

for each part ,give either a proof or a counterexample to justify your answer.

a)If is reflexive , must be reflexive ?

I was able to prove this part. I was just playing with the examples which

satisfy the above theorem.

Let

and lets define

clearly is reflexive. So S would be

is correct ? I included since , in the condition for

, we have ,

which can be written as an implication,

so if

then the antecedent is true always , as can be seen from the truth table of the

implication. But I couldn't include the ordered pairs like

in because if we let

and

then is true but is false

Which means that, in the implication , antecedent is true and the consequent is false,

which according to the truth table is false. So

doesn't satisfy the condition for the inclusion in .

is my reasoning correct ? If so, then we can see that is also reflexive,

which is what the above theorem asserts

I see one problem the way symbol for the power set P is displayed. I have used

\mathscr{P} to display slanted P, but it just displays ordinary . Is maths package **mathrsfs** installed on this forum ?

thanks

Re: problem about reflexive relation

Quote:

Originally Posted by

**issacnewton** is my reasoning correct ?

Yes,it is

Quote:

If so, then we can see that

is also reflexive, which is what the above theorem asserts

Right.

Quote:

I see one problem the way symbol for the power set P is displayed. I have used \mathscr{P} to display slanted P, but it just displays ordinary

. Is maths package

**mathrsfs** installed on this forum ?

With \mathcal{P}(A) we get .

Re: problem about reflexive relation

ola fernando

thanks for verifying.

adios

Re: problem about reflexive relation

Quote:

Originally Posted by

**issacnewton** ola fernando

Ola in Spanish means *wave* and hola means *hello*. :)

Quote:

thanks for verifying.

You are welcome!. :)

Re: problem about reflexive relation

Quote:

Originally Posted by

**issacnewton** is my reasoning correct ?

Of course, it's not a proof of the statement, it's just one example.

Re: problem about reflexive relation

Quote:

Originally Posted by

**emakarov** Of course, it's not a proof of the statement, it's just one example.

Of course, but ** isaacnewton** said "if so we can see that S is reflexive" i.e. he didn't say "my example proves S is reflexive".