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.