1. Hello! Merry Christmas and a Happy New Year.

I got one problem.

Alpha is relation on A x A, where A is set.

I need to prove:

alpha is symmetric and antisymmetric if and only if alpha is subset of {(x,x)|x is in A}

I tried to prove it this way.

Let x,y is in the set A.

Alpha is symetric:

If (x,y) is in alpha then (y,x) is in alpha.

Alpha is antisymetric:

If (x,y) is in alpha and (y,x) is in alpha then x=y

So I need to prove:

(P -> Q) ^ ((P ^ Q) -> S) <=> (P -> S)

where p denotes - (x,y) is in alpha
q denotes - (y,x) is in alpha
s denotes - x=y

But when I construct truth table (it is not tautology) it falls on:
P- true
Q-false
S-true

Is the theorem not correct or I did some mistake?

2.

3. I would skip the logic problem and just use a proof by contradiction. Assume that (x,y), for some x not queal to y, is in A.

4. Originally Posted by MagiMaster
I would skip the logic problem and just use a proof by contradiction. Assume that (x,y), for some x not queal to y, is in A.
Thank you.

Yes, I know that I could solve it that way.

I solved it that way, but I am interested in this way (because it is straightforward).

Could you possibly tell me what is the error?

P.S I also tried several other problems using that method but nothing, always false cases.

5. Magi, anybody?

6. Implication and truth tables always throws me a bit. The truth table for implication is a bit misleading. One of the true entries should really read something more like N/A, but since it's supposed to be either true or false, we say it's vacuously true.

In this particular case, you have P->Q is false but P and S are both true. I'm not sure this is particularly instructive.

In generaly, when you're using implications, I think you should either get rid of the implications (transform them in to something else), or use some method of proof other than a truth table. (You can still use boolean logic, but skip the truth table.)

 Bookmarks
##### Bookmarks
 Posting Permissions
 You may not post new threads You may not post replies You may not post attachments You may not edit your posts   BB code is On Smilies are On [IMG] code is On [VIDEO] code is On HTML code is Off Trackbacks are Off Pingbacks are Off Refbacks are On Terms of Use Agreement