set theory difficult question for the next few days in the univercity

[noam60]

Hello everyone.

I have tried those two questions using logical identities and I could not solve it.
I will really be glad for you help and I will very appreciate it.
I have two questions in discrete math.
It's for the next few days.


1) (? △ ?) ∩ ? = (A∩ C) △ (B ∩ C)
2) for I = {2,3,4}. Let the sets Ai = {i, 2i}, Bi = {i, i+1}
a) What are the elements of the set ∩i∈ I(Ai△Bi)?
b) What are the elements of the set

Thanks.

 

[Dr.Peterson]

Please show us whatever you are able to do here, so we can see what help you need. Also tell us what "logical identities" you have to work with.

I presume △ means the symmetric difference; do you know what that means? Problem 2 gives you practice in evaluating it. I think you intend i∈ I to be a subscript to the intersection, meaning "the intersection of all three sets Ai△Bi ". Write out what those are.

For problem 1, are you supposed to prove this? Please state the problem completely, and show some work (which might just be experimenting with particular sets or a Venn diagram, if you aren't ready for a proof).

 

[pka]

1) (? △ ?) ∩ ? = (A∩ C) △ (B ∩ C)
2) for I = {2,3,4}. Let the sets Ai = {i, 2i}, Bi = {i, i+1}
a) What are the elements of the set ∩i∈ I(Ai△Bi)?
b) What are the elements of the set

View attachment 15258 This is a diagram of symmetric difference. Do you see where the name may have come from?
\(\displaystyle x\in A\Delta B\) means that \(\displaystyle x\in(A\cap B^c)\vee x\in(A^c\cap B)\)

So \(\displaystyle x\in(A\Delta B)\cap C\\x\in[(A\cap B^c)\cup(A^c\cap B)]\cap C\\x\in([A\cap C]\cap B^c)\cup([B\cap C]\cap A^c)\\x\in(A\cap C)\Delta(B\cap C)\)

 

[Steven G]

pka, the attachment is not opening (at least for me)

 

[pka]

\(\displaystyle A\Delta B\)