MathDB

Let

a

and

b

be natural numbers. We consider the set

M

of the points of the plane with an integer

x

-coordinate from

1

a

and integer

y

-coordinate from

1

b

. For two points

P = (x, y)

and

Q = (\tilde x, \tilde y)

in M we write

P\le Q

x\le \tilde x

and

y \le \tilde y

, we say

P

is less than

Q

when

P\le Q

and

P \ne Q

. A subset

S

M

is now called cute if for every point

P \in S

it also contains all smaller points. From an arbitrary subset

S

M

we can now create new subsets in four ways to construct: (a) the complement

K (S) = \overline{S}

, (b) the subset

\min (S)

of its minima, i.e. those points for which there is no smaller in

S

occurs, (c) the cute set

P (S)

of all those points in M that are less than or equal to some point are from

S

, (d) you do all these things one after the other and get a set

Z (S) = P (\min (K (S)))

. Let

S

be cute. Prove that

\underset{a+b\,\, times\,\, Z}{Z(Z(...(Z(S))...))=S}

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