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Prove that the intersection of any number of rings of sets is
again a ring of sets.

______

Prove that the intersection of any number of s-rings of sets is
again a s-ring of sets.

_________

Let X be a nonempty set. Let W c P(X). What is "the ring
generated by W"?

The smallest ring containing W.

Such a ring exists.

It is the intersection of all rings containing W.

We denote it R(W).

Such a ring exists.

It is the intersection of all rings containing W.

We denote it R(W).

Let X be a nonempty set. Let W c P(X). What is "the s-ring
generated by W"?

The smallest s-ring containing W.

Such a s-ring exists.

It is the intersection of all s-rings containing W.

We denote it S(W).

Such a s-ring exists.

It is the intersection of all s-rings containing W.

We denote it S(W).

What is an algebra of sets? Give a minimal definition.

Let X be a set. Let R c P(X).

R is an algebra of sets

iff

(1) X :- R

(2) A,B :- R => A u B :- R

(3) B :- R => X \ B :- R

R is an algebra of sets

iff

(1) X :- R

(2) A,B :- R => A u B :- R

(3) B :- R => X \ B :- R

What is a s-algebra of sets?

Let X be a set. Let W c P(X).

W is a s-algebra of sets

iff

(1) X :- W

(2) /\n:-|N A[n]:-W |=> U(n:-|N)_A[n] :- W

(3) B:-W => X\B:-W

W is a s-algebra of sets

iff

(1) X :- W

(2) /\n:-|N A[n]:-W |=> U(n:-|N)_A[n] :- W

(3) B:-W => X\B:-W

Consider a collection of algebras of subsets of X.

X belongs to all the algebras.

Prove that the intersection of this collection is an algebra of subsets of X.

X belongs to all the algebras.

Prove that the intersection of this collection is an algebra of subsets of X.

page 33 in 1st measure

Consider a collection of s-algebras of subsets of X.

X belongs to all the s-algebras.

Prove that the intersection of this collection is a s-algebra of subsets of X.

X belongs to all the s-algebras.

Prove that the intersection of this collection is a s-algebra of subsets of X.

page 34 in first measure

Let X be a set. Let W c P(X).

What is "the algebra generated by W"?

What is "the algebra generated by W"?

The smallest algebra containing W.

Such an algebra exists.

It is the intersection of all algebras containing W.

We denote it K(W).

Such an algebra exists.

It is the intersection of all algebras containing W.

We denote it K(W).

Let X be a set. Let W c P(X).

What is "the s-algebra generated by W"?

What is "the s-algebra generated by W"?

The smallest s-algebra containing W.

Such a s-algebra exists.

It is the intersection of all s-algebras containing W.

We denote it d(W).

Such a s-algebra exists.

It is the intersection of all s-algebras containing W.

We denote it d(W).