Liked on YouTube: How to show/prove that if A and B are sets, then (A ∩ B) ∪ (A ∩ B') = A

How to show/prove that if A and B are sets, then (A ∩ B) ∪ (A ∩ B') = A
Problem 19.b in Section 2.2, Set Operations, Discrete Mathematics, 7th Ed Show that if A and B are sets, then (A ∩ B) ∪ (A ∩ B') = A //*Other books have it as "show (A ∩ B) ∪ (A - B) = A."*// //*(A intersect B) union (A intersect B') = A*// Solution Key definitions: A ⊂ B = {x : x ∈ A → x ∈ B} A ∩ B = {x : x ∈ A ∧ x ∈ B} A ∩ B' = A - B = {x : x ∈ A ∧ x ∉ B} A ∪ B = {x : x ∈ A ∨ x ∈ B} To show that (A ∩ B) ∪ (A ∩ B') = A, we need to show 2 things: 1. (A ∩ B) ∪ (A ∩ B') ⊂ A 2. A ⊂ (A ∩ B) ∪ (A ∩ B') //*Basically, you have 2 conditionals in a form "If P then Q" that you have to prove. To prove such conditional, you suppose that P is true, and you show that Q is true using facts about P.*// 1. Claim: (A ∩ B) ∪ (A ∩ B') ⊂ A. Proof: Suppose that x ∈ (A ∩ B) ∪ (A ∩ B'). Then, either x ∈ (A ∩ B) or x ∈ (A ∩ B'). So, either x ∈ A and x ∈ B or x ∈ A and x ∉ B. In either case, x ∈ A; thus, (A ∩ B) ∪ (A ∩ B') ⊂ A. 2. Claim: A ⊂ (A ∩ B) ∪ (A ∩ B') Proof: Suppose that x ∈ A. Then either x ∈ B or x ∉ B. If x ∈ B, then x ∈ A and x ∈ B; thus, x ∈ (A ∩ B). If x ∉ B, then x ∈ A and x ∉ B; thus, x ∈ (A ∩ B'). So, either x ∈ (A ∩ B) or x ∈ (A ∩ B'); that is, x ∈ (A ∩ B) ∪ (A ∩ B'). Therefore, A ⊂ (A ∩ B) ∪ (A ∩ B'). Q.E.D. Source(s): Grossman, Jerrold W., and Kenneth H. Rosen. "2.2 Set Operations." Student's Solutions Guide to Accompany Discrete Mathematics and Its Applications. 7th ed. New York: McGraw-Hill, 2012. 55. Print.
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Liked on YouTube: Ultimate Royce Gracie: Final Fight

Ultimate Royce Gracie: Final Fight
Every mixed martial artist in the world owes Royce Gracie, the original and three-time UFC tournament champion, a debt of gratitude. A true pioneer and UFC icon. Watch Royce's last fight against Kimo Leopoldo at UFC 3, then watch the full collection on UFC FIGHT PASS: http://ift.tt/1EVtWzU
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Attack of the Bear

Lion Vs Buffalo: Lions Attempt To Attack Buffalo Backfires

Whoa! Buffalo put up a fight and defend their own.

Caught On Camera: Leopard Runs Off With GoPro

Cold Winter Dawn

By Ramin Melikov

Not an angel, not a demon,
Just an earthly traveler,
Back from traveling the world,
Here to see your eyes.
Tired, cold, hungry and thirsty,
Desperate for nourishment,
Sleepless on this winter dawn,
I’m here scraping ice.

Happy birth day, been a while,
Here’s a card and flowers.
I drew the card myself, you know,
Grew flowers on my own.
There’s a poem in the card,
A short story about us.
I never wrote you one like that,
It’s different than before.

This poem is not like one
I wrote after our first kiss,
It’s different than the one
I wrote after our first love.
This poem’s different than all
The breakup poems I wrote,
It’s even different than the one
When you left me for good.

This poem’s pointless,
I’ll not read it aloud,
I’ll place it here on the stone,
Along with ring and flowers.
I’ll finish scraping this thick ice
That’s covering your eyes.
And then I’ll sit here ‘till dusk,
Burning all these poems.

Rebirth

By Ramin Melikov

Yesterday, I was trapped
In jaws of a lion,
Crying for help.
I died.
I died but I’m born again.
And tonight, I’m the lion
And you are my game.

But lucky for you,
My passion is rushing
To free you.
But you don’t wish to be freed.
Your eyes dare to test me,
You’re not even fighting.
You wish to be slayed.

I’m about to unleash
My gargantuan hunger
On this body of yours.
You are my possession.
I own you.
I’m on you.
I will do all that I want.

But first feel my roar.
Shiver now!
Your soul’s hour has come.
Enough of this foreplay.
I’m starving.
I will share you with no one.
You’re all will be mine.

Your cries will be loudest.
Your pain will be endless.
Your blood will be rushing.
Your eyes will be rolling.
“Oh God!” you’ll be screaming,
And thinking you’re dreaming,
And then begging for more.