## Discrete Mathematics : Proof : Question 1

1. The problem statement, all variables and given/known data

Question 1 :

a) Use Venn diagrams to determine whether or not, for all subnets A,B and C of a universal set U, (A-B) ∪ C = (A∪C) - (A∩B)
b) If the statement appears to hold, give a proof, if not, give a counter example.

2. Relevant equations

(A-B) ∪ C = (A∪C) - (A∩B)

*there are no other variables given
*no other values are known
*this question relates to the proof

3. The attempt at a solution

a) I have drawn the Venn diagrams, which does not reflect that they equate to each other, so they are not equal.
b) The counter example is the one I am struggling with, so i will explain how i did it, and basically just adapted an answer from my text book :

Attempt to prove with counter example :
------------------------------------------------------------

Let : A = {1;2}
Let : B = {2;3}
Let : C = {1;4}

Left hand : (A-B) ∪ C :

(A-B) = = {1;2} - {2;3} = {1;3}
(A-B) ∪ C = {1;3} ∪ C = {1;3} ∪ {1;4} = {1;3;4}

(A-B) ∪ C = {1;3;4}

Now to find out what the right hand side is :

(A∪C) - (A∩B) :

(A∪C) = {1;2}∪{1;4} = {1;2;4}
(A∩B) = {1;2}∩{2;3} = {2}
(A∪C) - (A∩B) = {1;4}

Thus :

(A-B) ∪ C ≠ (A∪C) - (A∩B)

-----------------------------------------------------------------------------

Please let me know if this is right, or where i can improve, this is something new to me, and i still need to work on this alot.

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 Your steps are not correct. A - B is the difference, so A - B = {1,2} - {2,3} = {1}. You are removing elements in AnB from A. So, what you have is not a counter example. Try again :)
 Your steps are not correct. A - B is the difference, so A - B = {1,2} - {2,3} = {1}. You are removing elements in AnB from A. So, what you have is not a counter example. Try again :)

## Discrete Mathematics : Proof : Question 1

Thanks,

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Left hand : (A-B) ∪ C :

(A-B) = = {1;2} - {2;3} = {1;3}
(A-B) ∪ C = {1;3} ∪ C = {1;3} ∪ {1;4} = {1;3;4}

(A-B) ∪ C = {1;3;4}
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should be :

=;=;=;=;=;=;=;=;=;=;=;=;=;=;=;=;=;=;=;=;=;=;=;=;

Left hand : (A-B) ∪ C :

(A-B) = = {1;2} - {2;3} = {1}
(A-B) ∪ C = {1} ∪ C = {1} ∪ {1;4} = {1;4}

(A-B) ∪ C = {1;4}

........

So that means that they are in fact the same.... (i did not see that from my Venn diagram).

I will quickly draw the venn diagrams again, is there a way i can show you the venn diagrams on this forum ?

 Ok, i have managed to upload a Venn diagram on these 2. Please let me know why my venn diagram does not reflect the calculation.. Attached Thumbnails
 Recognitions: Homework Help Ingenious! Your Venn diagrams are correct. If you find 3 sets which obey the original statement it can be still false. Find a counter-example. What about the sets in the attachment? ehild Attached Thumbnails
 Yes - if A n B n C is not empty, the proposition is false.
 hi Ehild, Thank you for the feedback, yes i can see the difference, only if a place a value in every field :) The example i used had 'empty fields' - which does not point out the difference. So in the process of answering this one (Correct me if i am wrong) : 1. Draw the venn diagrams 2. Put a value in every section (piece) of the venn diagram 3. Then do the calculations - due to the fields of the venn diagram , when writing out the proof it will be clear that the two are not the same Just writing out my proof, if you can confirm, i will really appreciate it : A = {1;2;6;7} B = {2;3;5;7} C = {4;5;6;7} (A-B) ∪ C = {1;2;6;7} - {2;3;5;7} ∪ C = {1;6} ∪ C = {1;6} ∪ {4;5;6;7} = {1;4;5;6;7} Right Hand : ∪ ∩ (A∪C) - (A∩ B) (A∪C) : {1;2;6;7) ∪ {4;5;6;7} = {1;2;4;5;6;7} (A∩B) : {1;2;6;7} ∩ {2;3;5;7} = {2;7} (A∪C) - (A∩ B) = {1;4;5;6} and Thus : (A-B) ∪ C ≠ (A∪C) - (A∩B)
 Nice, that works. A simpler example might have been: A = B = C = {1}. :)
 Who, Ehild, thanks alot for your assistance, appreciated!!