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A,B c X

Express differently

X \ (A \ B) = ???

Express differently

X \ (A \ B) = ???

X \ (A \ B) = (X \ A) u B

A,B c X

Express differently

(X \ A) u B = ???

Express differently

(X \ A) u B = ???

(X \ A) u B = X \ (A \ B)

Let X be a nonempty set. Let W be a collection of finite
subsets of X. What kind of collection is W?

W is an ideal.

W is a ring of sets.

W is a ring of sets.

Let X be a nonempty set. Let M c X. Let W be the collection of
all A c X such that MnA is finite or M\A is finite. What kind
of collection is W?

It is an algebra of sets.

PROOF:

A,B:-W

If MnA and MnB are both finite, then Mn(AuB) is finite, hence AuB:-W.

If M\A is finite, then M\(AuB) c M\A, hence M\(AuB) is finite, hence AuB:-W.

PROOF:

A,B:-W

If MnA and MnB are both finite, then Mn(AuB) is finite, hence AuB:-W.

If M\A is finite, then M\(AuB) c M\A, hence M\(AuB) is finite, hence AuB:-W.

Let X be a nonempty set. Let W be the collection of all
countable subsets of X. What kind of collection is W?

It is a s-ring of sets.

It is a s-ideal.

It is a s-ideal.

Let X be a nonempty set. Let M c X. Let W be the collection of
all A c X such that MnA is countable or M\A is countable. What
kind of collection is W?

It is a s-algebra of sets.

How do I choose to understand the word "countable" ?

countable = empty or finite or equinumerous with the set of
natural numbers

Is it true that every countable union of sets in a ring may be
written as a disjoint countable union of sets in the ring?

U(n:-|N) A[n] = \\*//(n:-|N) [ A[n] \ (U(k=1 to k=n-1) A[k]) ]

where A[n] = O

page 41 in 1st measure

where A[n] = O

page 41 in 1st measure

(1) h : |R -> |R is diffable

(2) h(0) = 0

(3) /\x:-|R h'(x) + h(x)*cos(x) = cos(x)

Find the set of all such functions.

(2) h(0) = 0

(3) /\x:-|R h'(x) + h(x)*cos(x) = cos(x)

Find the set of all such functions.

There is one and only one such function.

h(x) = 1 - exp(-sin(x))

hint: multiply both sides of (3) by exp(sin(x))

h(x) = 1 - exp(-sin(x))

hint: multiply both sides of (3) by exp(sin(x))

Let A[n] be a sequence of sets.

Express its union as the union of an increasing sequence of sets.

Express its union as the union of an increasing sequence of sets.

U{A[n]} = U(n:-N) [ U(k=1 to k=n) A[k] ]