How to Prove it book help with ch 1.4 operation on sets problem (symmetric diff)

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The discussion focuses on verifying the identity (A ∩ B) Δ C = (A Δ C) Δ (A \ B) using Venn diagrams. Participants express confusion over their attempts to represent the left-hand side correctly, particularly regarding the intersections and unions involved in symmetric differences. Several diagrams are shared, with feedback indicating that initial attempts are incorrect and need adjustments to accurately depict the sets. The importance of clearly illustrating the relationships between the sets is emphasized, and users are encouraged to take breaks if they feel overwhelmed. Ultimately, the conversation highlights the challenges of visualizing set operations and the collaborative effort to clarify the problem.
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Homework Statement



Use any method you wish to verify the following identity:

(A \cap B) Δ C = ( A Δ C) Δ (A \ B)

Homework Equations



A Δ B = (A \ B) \cup (B \ A) = (A \cup B) \ (A \cap B)

The Attempt at a Solution



http://img17.imageshack.us/img17/48/14question14b.jpg


I think my attempt at the left hand side Venn diagram ((A \cap B) Δ C) is incorrect as the right hand side states that x \notin B, so I am left very confused :(
 
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Gerlan_au said:

Homework Statement



Use any method you wish to verify the following identity:

(A \cap B) Δ C = ( A Δ C) Δ (A \ B)

Homework Equations



Definition of A Δ B :

A Δ B = (A \ B) \cup (B \ A) = (A \cup B) \ (A \cap B)

The Attempt at a Solution



[ IMG]http://i.imgur.com/cmOJv.jpg[/PLAIN]

I think my attempt at the left hand side Venn diagram ((A \cap B) Δ C) is incorrect as the right hand side states that x \notin B, so I am left very confused :(
You should really reduce the size of that image you posted.

I take it that the Δ symbol stands for the symmetric difference of two sets.


The figure you show seems nonsensical .
 
Thank you for replying, resized the image, and yes the Δ stands for symmetric difference, I also think that my original attempt at the solutions is incorrect, could you possibly point me to the right direction?

The book itself did not have any solved solutions for similar problems so I cannot confirm if my solution is correct or not and it is bugging the hell out of me >< (obviously the solution is not correct :P)
 
Your very first diagram for (A ∩ B) Δ C is wrong. Try drawing (A ∩ B) first. The second one for (A Δ C) looks ok. The third one for (A \ B) is also wrong. If you are going to put this all together with Venn diagrams you need to correct some of the parts.
 
Thanks for the reply sir! :)

I have tried to correct my diagram and came up with this, is that in the right direction? (I think for (A ∩ B) Δ C I should remove the intersection between B and C )

http://img152.imageshack.us/img152/4692/14question14battempt2.jpg
 
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Gerlan_au said:
Thanks for the reply sir! :)

I have tried to correct my diagram and came up with this, is that in the right direction? (I think for (A ∩ B) Δ C I should remove the intersection between B and C )

http://img152.imageshack.us/img152/4692/14question14battempt2.jpg

The symmetric difference should be everything that is in (A ∩ B) or C but not in A ∩ B ∩ C. Right? You've still got the same diagram as in the original post. I think you need to remove some stuff and add some stuff.
 
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Hmmm, would that be correct then?

http://img89.imageshack.us/img89/9255/14question14battempt3.jpg

Thank you very much for taking the time to help me though! At the moment I've been working on the problems from that chapter for the better part of 5 hours and I do not think I can think straight anymore lol, so if my above correction is wrong I might take a break and return to that question tomorrow as my brain is already fried :(
 
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Gerlan_au said:
Hmmm, would that be correct then?

http://img89.imageshack.us/img89/9255/14question14battempt3.jpg

Thank you very much for taking the time to help me though! At the moment I've been working on the problems from that chapter for the better part of 5 hours and I do not think I can think straight anymore lol, so if my above correction is wrong I might take a break and return to that question tomorrow as my brain is already fried :(

That's right! Take a break or try to fix the third diagram. That's an easy one. Your choice.
 
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Alright so I tried to take a break from that problem but it is still in the back of my head and I think i may have found a solution!

http://img7.imageshack.us/img7/9255/14question14battempt3.jpg

Is that right?! Am I close?! or am I just going insane and have not noticed it hah!
 
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  • #10
Gerlan_au said:
Alright so I tried to take a break from that problem but it is still in the back of my head and I think i may have found a solution!

http://img7.imageshack.us/img7/9255/14question14battempt3.jpg

Is that right?! Am I close?! or am I just going insane and have not noticed it hah!

I'm not sure whether it's right or not. What's it supposed to be?
 
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  • #11
Gerlan_au said:
Alright so I tried to take a break from that problem but it is still in the back of my head and I think i may have found a solution!

http://img7.imageshack.us/img7/9255/14question14battempt3.jpg

Is that right?! Am I close?! or am I just going insane and have not noticed it hah!
I'm pretty sure that the question Dick was asking was, What is this the Venn of ?

Can't say it's right or that it's wrong without knowing what it refers to.

I'm sure Dick can write an expression which corresponds to this Venn diagram. On a good day, so can I.
 
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