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

  • Thread starter Gerlan_au
  • Start date
  • Tags
    Book Sets
In summary, the diagram for (A ∩ B) Δ C is wrong. I think you should try to draw (A ∩ B) first and correct it. The diagram for (A Δ C) looks ok. The diagram 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! :)
  • #1
Gerlan_au
5
0

Homework Statement



Use any method you wish to verify the following identity:

(A [itex]\cap[/itex] B) Δ C = ( A Δ C) Δ (A \ B)

Homework Equations



A Δ B = (A \ B) [itex]\cup[/itex] (B \ A) = (A [itex]\cup[/itex] B) \ (A [itex]\cap[/itex] 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 [itex]\cap[/itex] B) Δ C) is incorrect as the right hand side states that x [itex]\notin[/itex] B, so I am left very confused :(
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
Gerlan_au said:

Homework Statement



Use any method you wish to verify the following identity:

(A [itex]\cap[/itex] B) Δ C = ( A Δ C) Δ (A \ B)

Homework Equations



Definition of A Δ B :

A Δ B = (A \ B) [itex]\cup[/itex] (B \ A) = (A [itex]\cup[/itex] B) \ (A [itex]\cap[/itex] 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 [itex]\cap[/itex] B) Δ C) is incorrect as the right hand side states that x [itex]\notin[/itex] 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 .
 
  • #3
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)
 
  • #4
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.
 
  • #5
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
 
Last edited by a moderator:
  • #6
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.
 
Last edited by a moderator:
  • #7
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 :(
 
Last edited by a moderator:
  • #8
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.
 
Last edited by a moderator:
  • #9
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!
 
Last edited by a moderator:
  • #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?
 
Last edited by a moderator:
  • #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.
 
Last edited by a moderator:

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

1. What is the "symmetric difference" operation on sets?

The symmetric difference of two sets A and B is defined as the set of elements that are in either A or B, but not in both. It is denoted by A △ B.

2. How do I prove that A △ B = (A ∪ B) \ (A ∩ B)?

To prove this, you can use the definition of symmetric difference and set identities. First, show that any element in A △ B is also in (A ∪ B) \ (A ∩ B). Then, show that any element in (A ∪ B) \ (A ∩ B) is also in A △ B. This will prove that the two sets are equal.

3. Can I use a Venn diagram to visualize the symmetric difference operation?

Yes, a Venn diagram can be a helpful tool to visualize the symmetric difference operation. The symmetric difference can be represented as the shaded area in the middle of two overlapping circles, where each circle represents one of the sets A and B.

4. How is the symmetric difference operation related to the union and intersection of sets?

The symmetric difference operation is closely related to the union and intersection of sets. In fact, it can be expressed in terms of these operations as A △ B = (A ∪ B) \ (A ∩ B). This means that the symmetric difference is a combination of the union and intersection operations.

5. Can the symmetric difference operation be extended to more than two sets?

Yes, the symmetric difference operation can be extended to any number of sets. For example, the symmetric difference of three sets A, B, and C can be written as A △ B △ C. The definition remains the same - it is the set of elements that are in an odd number of the given sets.

Similar threads

How Can I Prove the Set Identity (A ∩ B) △ C = (A △ C) △ (A \ B)?
    • Precalculus Mathematics Homework Help
Replies
16
Views
2K
Solve Set Theory Homework: Right Hand Side of "Or
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Venn Diagram for set operations
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Set Theory - Proving Contrapositive
    • Precalculus Mathematics Homework Help
Replies
6
Views
3K
How many surjective functions are there from {1,2,...,n} to {a,b,c,d}?
    • Calculus and Beyond Homework Help
Replies
2
Views
2K
Simple Set Theory Proofs to Proving Set Identities - Homework Help
    • Calculus and Beyond Homework Help
Replies
10
Views
1K
How Do De Morgan's Laws Apply to Set Theory Proofs?
    • Calculus and Beyond Homework Help
Replies
13
Views
2K
Euclidean and non Euclidean geometries problems
    • Precalculus Mathematics Homework Help
Replies
4
Views
3K
Validating Set Equivalencies: A Logical Approach
    • Calculus and Beyond Homework Help
Replies
2
Views
2K
Determine if true or false F(A-B) = F(A) -F(B)
    • Calculus and Beyond Homework Help
Replies
10
Views
10K

Hot Threads

Recent Insights

Back
Top