What's everyone up to right now?

[c172]

Reading a Conservative Tribune article on a gymnast that was forced to change her floor music at the last minute. It started with a few bars of "Dixie's Land", which offended some folks.

[Cecelia]

Dreading going back to work tomorrow. Summer vacation over =/ My kids still have a week off at least. And I don't have to start grading homework until the next week.

[SteelCurtain]

Back in Tennessee.

Back to reality. It's oddly good to be home.

[Little lunch]

Feeling awful.  Seven days since I ran out of Zoloft.  Hopefully it will be in the mail tomorrow.  Antidepressant withdrawal is not fun...🙁

I'm on Zoloft also.
I find that a week without it will sought of then take another week to kick in properly.
A lot of changes can happen in that time and who knows where, mentally, you'll be by then.
It some ways I sought of like it because I get angrier and more powerful and dangerous, and that can be, weirdly enough, a good feeling.
But if I use that power I'll eventually tire out and then life becomes pointless.
I have to rely on memories of why I liked being alive in the first place.
Having been through it a lot of times helps. You get more experienced at handling it.
Same goes for the people around you, the ones you have left.
Anyway, hang in there. You're not alone. :-)

[Jackalope]

Feeling awful.  Seven days since I ran out of Zoloft.  Hopefully it will be in the mail tomorrow.  Antidepressant withdrawal is not fun...🙁

I'm sorry you're going through that. If I don't regularly take mine (Lamictal) I have bad mood swings. Hope you get them soon.

ETA: Them being your meds, not mood swings.

[Kernel Sohcahtoa]

I've been playing around with some proofs. In particular, I enjoyed proving the following proposition:

Prove that (ℤ x ℕ)∩(ℕ x ℤ)=(ℕ x ℕ)

I actually used the technique of transforming the left side into the right side via a series of equalities using set notation.  However, I was considering the standard approach of showing that the first set is a subset of the second set and the second set is a subset of the first set.  Hence, this is the technique used in the proof below.

Please note that I've put the proof in hide tags, so that any interested persons can have a go at the proof for themselves.

With that said, here are some definitions:

Definition 1: The Cartesian product of two sets A and B is another set, denoted as A x B and defined as A x B={(x,y): x is in A and y is in B}

Definition 2: The intersection of a set A and a set B is the set A∩B= {x: x is in A and x is in B}.

Definition 3: A set S is a subset of a set T (written as S⊆T) if and only if for every element x in S, x is in T.

Definition 4: Two sets S and T are equal (written S=T) if and only if S is a subset of T and T is a subset of S. 

Definition 5: The set of integers is denoted as ℤ={...,-2,-1,0,1,2,...}. In other words, the set of integers equals the set containing zero and every whole number on the real number line.

Definition 6: The set of natural numbers is denoted as ℕ={1,2,3,....}.  In other words, the set of natural numbers is the set of positive integers.    


Proof.

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Now, in order to show that (ℤ x ℕ)∩(ℕ x ℤ)⊆(ℕ x ℕ), suppose that (x,y) is in (ℤ x ℕ)∩(ℕ x ℤ).  Via the definition of intersection, (x,y) is in (ℤ x ℕ) and (x,y) is in (ℕ x ℤ).  Via the definition of Cartesian product, x is in ℤ and y is in ℕ and x is in ℕ and y is in ℤ.  In other words, we have that, x is in ℕ and y is in ℕ and x is in ℤ and y is in ℤ. Via the definition of Cartesian product, (x,y) is in (ℕ x ℕ) and (x,y) is in (ℤ x ℤ).  Via the definition of intersection, (x,y) is in (ℕ x ℕ)∩(ℤ x ℤ).  Now, ℕ x ℕ is the set of all points on the Cartesian plane whose coordinates are both positive integers. (ℤ x ℤ) is the set of all points on the Cartesian plane whose coordinates span the entire set of integers: in other words, every possible coordinate pair of integers is accounted for on the Cartesian plane which means that (ℕ x ℕ)⊆(ℤ x ℤ).  Thus, it follows that the intersection of (ℕ x ℕ) and (ℤ x ℤ) is (ℕ x ℕ) or (ℕ x ℕ)∩(ℤ x ℤ)=(ℕ x ℕ). Consequently, (x,y) is in (ℕ x ℕ).  Hence, (x,y) is in (ℤ x ℕ)∩(ℕ x ℤ) implies (x,y) is in (ℕ x ℕ) which means that (ℤ x ℕ)∩(ℕ x ℤ)⊆(ℕ x ℕ).


To establish that (ℕ x ℕ)⊆(ℤ x ℕ)∩(ℕ x ℤ), suppose that (x,y) is in (ℕ x ℕ).  By the definition of Cartesian product, x is in ℕ and y is in ℕ.  Now, we know that ℕ⊆ℤ. Via the definition of subset, if x is in ℕ, then x is in ℤ.  Likewise, if y is in ℕ, then y is in ℤ. Thus, we have that, x is in ℤ and y is in ℕ and x is in ℕ and y is in ℤ.  Via the definition of Cartesian product, (x,y) is in (ℤ x ℕ) and (x,y) is in (ℕ x ℤ).  Via the definition of intersection, we have that (x,y) is in (ℤ x ℕ)∩(ℕ x ℤ).  Hence, (x,y) is in (ℕ x ℕ) implies that (x,y) is in (ℤ x ℕ)∩(ℕ x ℤ) which means that (ℕ x ℕ)⊆(ℤ x ℕ)∩(ℕ x ℤ). 


The previous two paragraphs established that (ℤ x ℕ)∩(ℕ x ℤ)⊆(ℕ x ℕ) and (ℕ x ℕ)⊆(ℤ x ℕ)∩(ℕ x ℤ).  Hence, it follows that (ℤ x ℕ)∩(ℕ x ℤ)=(ℕ x ℕ), which completes the proof.

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[Longhorn]

Listening to hateful songs because I'm mad >:|

May I suggest some Hocico or Terminal choice?

whoops it's been six hours since then, but next time you are filled with anger...

Thanks! I'll check them out

[Kernel Sohcahtoa]

I'm watching the Star Trek episode "The Ultimate Computer."

[The Valkyrie]

Bored.

I'm sitting in a tiny room at a NZ navy base with no TV and a couple of books.

About 5 minutes' walk from me is a warship and no one will let me borrow it!

[ignoramus]

Beccs, you can always sit on the turret and start singing like Cher!