Existentialism Mathematics #11

March 25, 2010

Notes on Existentialism Mathematics by dr. Algirdas Javtokas (New Frontiers in Mathematics Series).

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Lemma 1. 1 1 1 0 1 01 0 1 0 1 0 1 0 0 1 1 1 1 01 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 00 1 0 1 0 1 0 1 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 01 0 0 0 0 1 11 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.

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0 1 1 0 1 010 1 1 0 1 0 1 0 1 0 1 0 1 0 1 010 1 1 1 1 1 1 0 0 0 0 0 01 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 011 0 1 0 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 0

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Proof  1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 11 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0

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0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0

11100 \displaystyle\bigcirc 0000 \displaystyle\bigcirc 000

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Lemma 2. 0 1 1 0 1 01 0 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 0 1 1 1 0 00 1 0 1 0 1 0.

\displaystyle\bigodot 01110 \displaystyle\bigcirc 00 \displaystyle\bigcirc 0110 = 000

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Proof  1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 11 0 1 0 1 0 0 1 1 0 1 01 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1

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0 1 01 01 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 10 0 0 0 1

00111110 \displaystyle\bigoplus 001

0 1 1 1 1 0 00 1 0 1 0 1 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 0 11 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 01 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1

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Theorem. 0 1 01 0 1 0 1 0 0 0 0 1 11 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 00 1 0 1

000 \displaystyle\bigcup 000 \displaystyle\oplus 000 \displaystyle\bigcap 000

1 1 0 1 01 0 1 0 1 0 0 0 0 11 1 1 1 1 1 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1

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Proof  0 1 01 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 00 0 0 0 0 1 0 1 0 0 0 0 1 11 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 0 00 1 0 1

0000001000 \displaystyle\bigcirc 000 \displaystyle\subset 000 \displaystyle\subset 000

0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0

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0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 1 1 1 1 1 1 10 1 0 1 0 0 1 1 0 1 01 0 1 0 1 01 0 1 0 1 01 0 10 1 0 10 1 0 10 1 0 1 00 1 1 1 1 1 0 0 0 0 0 0 0

010 \displaystyle\bigoplus 001110 = 000111 \displaystyle\bigotimes 010

0 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 0 00 1 0 1 0 1 0 0 1 1 0 11 01 0 1 0 0 0 1 1 1 1

\displaystyle\bigcup 010 \displaystyle\bigcup 00111110 \displaystyle\bigcup 0111100

0 1 1 0 1 1 1 1 1 1 1 1 1 1 10 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 1 1 1 11 1 1 1 1 0 0 0 0 0 0 1 1 1 1 11 00 0 0 0 1 1 1 1 1 1 0 1 1 1 00 1 1 1 0 0 1 11 0 1 1 0 1 1 1 00 1 1 0 0 01 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 10 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 11 1 1 11 0 1 0 1 0 01 1 1 1 0 0 1 1 1 1

0111110 \displaystyle\bigcirc 001110 \displaystyle\equiv 00110 \displaystyle\displaystyle\bigcap 000

0 1 01 0 1 1 11 1 1 1 1 1 1 1 0 0 0 0 1 1 11 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 1 1 1 1 1 1 1 1 1 101

1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 0 0 1 1 0 1 01 0 1 0 1 0 0 0 0 1 1 1 1 0 00 1 0 1 0 1 01 1 0 1 01 0 1 0 1 1 1 1 1 1 1 1 1 1 101

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