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Let A,B,C,D,E,F be non-empty sets.

Suppose that (AxC) u (BxD) = ExF.

Suppose that A n B = O.

What can we conclude?

Suppose that (AxC) u (BxD) = ExF.

Suppose that A n B = O.

What can we conclude?

C = D

Take any d:-D. Choose some b:-B. Now (b,d):-BxD. Hence (b,d):- ExF.

Choose some (a,c):-AxC. Now (a,c):-ExF. We conclude that (a,d):- ExF.

Hence (a,d):-AxC or (a,d):-BxD. But a:-A and AnB=O, h

Take any d:-D. Choose some b:-B. Now (b,d):-BxD. Hence (b,d):- ExF.

Choose some (a,c):-AxC. Now (a,c):-ExF. We conclude that (a,d):- ExF.

Hence (a,d):-AxC or (a,d):-BxD. But a:-A and AnB=O, h

Let A,B,C,D be non-empty sets.

Prove that if A=B or C=D, then

(A x C) u (B x D) = (A u B) x (C u D).

Prove that if A=B or C=D, then

(A x C) u (B x D) = (A u B) x (C u D).

___

Let J1 be a semi-ring in X, and J2 a semi-ring in Y.

Let u : J1 -> [0,oo] and y : J2 -> [0,oo] be additive.

Let J = { E x F : E:-J1 and F:-J2 }.

Let q : J -> |R*, q(E x F) = u(E) * y(F).

What can we conclude about q?

Let u : J1 -> [0,oo] and y : J2 -> [0,oo] be additive.

Let J = { E x F : E:-J1 and F:-J2 }.

Let q : J -> |R*, q(E x F) = u(E) * y(F).

What can we conclude about q?

q is additive

page 89 in 1st measure

page 89 in 1st measure

Let W be a ring of sets.

Let y : W -> |R* be additive.

Prove that for every A,B :- W

y(A u B) + y(A n B) = ???

Let y : W -> |R* be additive.

Prove that for every A,B :- W

y(A u B) + y(A n B) = ???

y(A u B) + y(A n B) = y(A) + y(B)

1) A u B = A u (B\A) disjointly

2) y(A u B) = y(A) + y(B\A)

3) y(A u B) + y(A n B) = y(A) + y(B\A) + y(B n A)

4) y(A u B) + y(A n B) = y(A) + y(B)

1) A u B = A u (B\A) disjointly

2) y(A u B) = y(A) + y(B\A)

3) y(A u B) + y(A n B) = y(A) + y(B\A) + y(B n A)

4) y(A u B) + y(A n B) = y(A) + y(B)

Let W be a ring of sets.

Let y : W -> |R* be additive.

Prove that for every A,B :- W

y(A) + y(B) = ???

Let y : W -> |R* be additive.

Prove that for every A,B :- W

y(A) + y(B) = ???

y(A) + y(B) = y(A u B) + y(A n B)

PROOF:

0) A u B = A u (B\A) disjointly

1) y(A u B) + y(A n B) = y(A) + y(B\A) + y(B n A)

2) y(A u B) + y(A n B) = y(A) + y(B)

PROOF:

0) A u B = A u (B\A) disjointly

1) y(A u B) + y(A n B) = y(A) + y(B\A) + y(B n A)

2) y(A u B) + y(A n B) = y(A) + y(B)

Let (T,>) be a directed set and let p:-T.

Let P = { t:-T : t > p }.

What can we say about (P,>) ?

Let P = { t:-T : t > p }.

What can we say about (P,>) ?

(P,>) is also a directed set

Let W c P(X). Let A,B c X, x :- X.

Suppose that x is W-adherent to A u B.

What can we conclude?

Suppose that x is W-adherent to A u B.

What can we conclude?

x is W-adherent to A or x is W-adherent to B

page 53 in gen top

page 53 in gen top

Let W c P(X), A,B c X.

Suppose that A,B are W-closed.

What can we conclude?

Suppose that A,B are W-closed.

What can we conclude?

1) A u B is W-closed

2) A n B is W-closed

page 54 in gen top

2) A n B is W-closed

page 54 in gen top

Let W c P(X), A,B c X.

Suppose that A,B are W-open.

What can we conclude?

Suppose that A,B are W-open.

What can we conclude?

A n B is W-open, page 55 in gen top

A u B is W-open, page 56 in gen top

A u B is W-open, page 56 in gen top

Let W c P(X). Let M c P(X) be a collection of W-closed sets.

What can we conclude?

What can we conclude?

//\\M is W-closed

page 55 in gen top

page 55 in gen top