Sets of sets question help

Prove that if an element is in the left hand side then it is in the right hand side so, the left hand side is contained in the right hand side, and then do the same thing the other way around. These questions are usually just looking at the definitions.

Reply 2

OP

Let x be an element of S

so x is an element of S1 OR x is an element of S2 because S = S1 u S2

So, U(S) = U(S1) OR U(S) = U(S2) therefore U(S) = U(S1) u U(S2)

Is this sufficient... I feel like this isn't right?

(edited 5 years ago)

Reply 3

OP

I'm not sure about the other way round

Reply 4

Original post by omarathon

Let x be an element of S

so x is an element of S1 OR x is an element of S2 because S = S1 u S2

So, U(S) = U(S1) OR U(S) = U(S2) therefore U(S) = U(S1) u U(S2)

Is this sufficient... I feel like this isn't right?

I think the U(S) notation is specific to the book you're using, so I don't know what it means, what does it mean?

If you want to be super pedantic you have to make sure the sets are not empty, but if its quite easy to check the result is true if s1 and s2 are empty, since the empty set is unique.

I don't think your last line correct. You are trying to prove that U(S) = U(S1) U (US2) but you have stated something that is not true in general,

A = B

or

A = C

does not imply that

A = B \cup C

where A B and C are sets

Reply 5

OP

let x be element of U(S1) u U(S2)

so x is an element of U(S1) or U(S2)

how do I go from that if x is an element of U(S1), x is an element of S1? Because the elements of U(S1) differ from the elements of S1.

Reply 6

OP

Original post by univ4464

I think the U(S) notation is specific to the book you're using, so I don't know what it means, what does it mean?

If you want to be super pedantic you have to make sure the sets are not empty, but if its quite easy to check the result is true if s1 and s2 are empty, since the empty set is unique.

I don't think your last line correct. You are trying to prove that U(S) = U(S1) U (US2) but you have stated something that is not true in general,

A = B

or

A = C

does not imply that

A = B \cup C

where A B and C are sets

The U notation is the union of all sets within a set, so if S = {{1,2,3}, {3,4,5}, {5,6,7}} then U(S) = {1,2,3,4,5,6,7}.

ok so I am not sure what to do if the last line isn't true

Reply 7

Original post by omarathon

The U notation is the union of all sets within a set, so if S = {{1,2,3}, {3,4,5}, {5,6,7}} then U(S) = {1,2,3,4,5,6,7}.

ok so I am not sure what to do if the last line isn't true

I think you could start by doing something like this

x \in \cup S \Rightarrow x \in A \in S_1 \text{ or } x \in B \in S_2

and then if it is in

A

then

x \in \cup S_1 \Rightarrow x \in (\cup S_1) \cup (\cup S_2)

and then the same thing for

B

Reply 8

OP

thanks but can you explain that if x belongs to S1 this means that x belongs to U(S1)? S1 is a set of sets and U(S1) is a set of elements which are in the sets embedded in S1, so S1 doesn't exist in U(S1) since U(S1) doesn't contain sets? I'm probably over-complicating it since it's obvious why if x belongs so a set Y then x also belongs to u(Y) but the capital U makes it strange. Also why do you introduce A and B, if they just translate to S1 and S2 respectively? Is this a convention?

(edited 5 years ago)

Reply 9