Set Theory Proof(Using Identities)

Click For Summary
The discussion revolves around solving a proof in set theory, specifically proving that A U B equals (A ∩ B') U (A' ∩ B) U (A ∩ B). The user seeks assistance after struggling for three hours. Participants confirm the use of standard set axioms like distributivity and DeMorgan's laws. The proof is simplified, ultimately leading to the conclusion that A U B is derived correctly. The user expresses satisfaction upon reaching the solution after extensive work.
ozymandris
Messages
5
Reaction score
0
Hi, I've been trying for 3 hours to solve this proof using identities. I can't seem to get it.

Can i get a little help please?

Prove: A U B = (A ∩ B') U (A' ∩ B) U (A ∩ B)

thanks
 
Physics news on Phys.org
welcome to pf!

hi ozymandris! welcome to pf! :smile:

do the easy bit first …

what is (A' ∩ B) U (A ∩ B) ? :wink:​
 
ozymandris said:
Hi, I've been trying for 3 hours to solve this proof using identities. I can't seem to get it.

Can i get a little help please?

Prove: A U B = (A ∩ B') U (A' ∩ B) U (A ∩ B)

thanks

Hey ozymandris and welcome to the forums.

What are you allowed to take for granted? Can you use normal set axioms like distributivity, DeMorgans laws, and so on?
 
chiro said:
Hey ozymandris and welcome to the forums.

What are you allowed to take for granted? Can you use normal set axioms like distributivity, DeMorgans laws, and so on?

(A' ∩ B) U (A ∩ B) = (A' U A) ∩ B


We're allowed all the normal set axioms, distributivity, deMorgan, etc etc
 
ozymandris said:
(A' ∩ B) U (A ∩ B) = (A' U A) ∩ B

yup! :smile:

and A' U A = … ? :wink:​
 
tiny-tim said:
yup! :smile:

and A' U A = … ? :wink:​

ok, i think I've gotten it. It took me nearly a page of workout. I'll post it when i have a minute to type it all in.
 
It should be only two lines, so you might want to see if you can shorten your reasoning.
 
tiny-tim said:
yup! :smile:

and A' U A = … ? :wink:​

A' U A = \varnothing
 
(A ∩ B') U (A' U A) ∩ B

(A ∩ B') U {Universal} ∩ B

(A ∩ B') U B if i distribute from this

(A U B ) ∩ (B' U B)

A U B
 
  • #10
:biggrin: Woohoo! :biggrin:​
 

Similar threads

Undergrad Countability of binary Strings
  • · Replies 18 ·
Replies
18
Views
3K
Undergrad Help understanding conditional expectation identity
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Is this conditional expectation identity true?
  • · Replies 1 ·
Replies
1
Views
2K
MHB Inequality of Cardinality of Sets
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Prove that points which are indistinguishable from 0 exist (using logic)
  • · Replies 17 ·
Replies
17
Views
2K
Undergrad Missing(?) rigor in proof involving countable union of countable sets
  • · Replies 10 ·
Replies
10
Views
2K
High School Two interesting number theory tricks
  • · Replies 3 ·
Replies
3
Views
769
MHB Proof of a set union and intersection
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Supremum proof & relation to Universal quantifier
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Is an algorithm for a proof required to halt?
  • · Replies 3 ·
Replies
3
Views
2K