What Does Unique Ring P Containing S Imply in Set Theory?

[woundedtiger4]

Question on Ring...Help Please!

Given any non-empty systems of sets S, there is a unique ring P containing S and contained in every ring containing S. The ring P is called the minimal ring generated by the system S & can be denoted as R(S).
Question: what does mean by "there is a unique ring P containing S", does it mean that P is in S ? if I am wrong then is P a maximal set of S?

 

[sunjin09]

Given any non-empty systems of sets S, there is a unique ring P containing S and contained in every ring containing S. The ring P is called the minimal ring generated by the system S & can be denoted as R(S).
Question: what does mean by "there is a unique ring P containing S", does it mean that P is in S ? if I am wrong then is P a maximal set of S?

It means that S$\subset$P, i.e., S is a set of sets, and P is a RING OF SETS, which contains every element of S as its element (and more, in general). This ring P is said to be generated by the set S, since it is created by adding (symmetric difference) and multiplying (intersection) elements of S, and collecting all the possible outcomes.

 

[woundedtiger4]

It means that S$\subset$P, i.e., S is a set of sets, and P is a RING OF SETS, which contains every element of S as its element (and more, in general). This ring P is said to be generated by the set S, since it is created by adding (symmetric difference) and multiplying (intersection) elements of S, and collecting all the possible outcomes.

Thanks, your answer is really helpful.
one more question, is it necessary that a system of sets is always a ring? Plus, I don't understand by what you said "This ring P is said to be generated by the set S, since it is created by adding (symmetric difference) and multiplying (intersection) elements of S, and collecting all the possible outcomes"?

 

[sunjin09]

Thanks, your answer is really helpful.
one more question, is it necessary that a system of sets is always a ring? Plus, I don't understand by what you said "This ring P is said to be generated by the set S, since it is created by adding (symmetric difference) and multiplying (intersection) elements of S, and collecting all the possible outcomes"?

Of course not, given a universal set X, the power set P(X) is a ring of sets (the two definitions of addition and multiplication of sets satisfy all the axioms of ring addition and multiplication, and P(X) is certainly closed under these compositions.) This biggest ring of sets on X may have subrings, which are closed under these compositions. An arbitrary system of sets aren't necessarily closed under these compositions, but can be supplemented with other necessary sets to make it closed, i.e., turn it into a subring of P(X). It is in this sense that an arbitrary system of set can generate a (sub)ring.

 

[blue_raver22]

I can provide a response to this content by explaining what a ring is and how it relates to the statement provided. A ring is a mathematical structure that consists of a set of elements and two operations, usually addition and multiplication, that satisfy certain properties. In this case, the set S is a non-empty system of sets, which means it is a collection of sets that may or may not have any common elements.

The statement is saying that given any non-empty system of sets S, there is a unique ring P that contains S. This means that P is a ring that contains all the elements of S and also satisfies the properties of a ring. P is not necessarily an element of S, but rather a separate ring that contains the elements of S.

P is not a maximal set of S, as it is not a subset of S. P is a ring that contains S and is contained in every other ring that also contains S. This makes P the minimal ring generated by the system S.

To understand this concept better, think of S as a starting point or a set of basic elements. P is then constructed using these basic elements and satisfies the properties of a ring. P is the smallest possible ring that can be constructed using the elements of S. Any other ring that contains S will also contain P.

I hope this explanation helps clarify the concept of a minimal ring generated by a system of sets. Please let me know if you have any further questions.