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( 2 0 points ) Uncountability / Diagonalization ( a ) At the Dawn of Time, an infinite sequence of Pennies from Heaven fell to
points Uncountability Diagonalization
a At the Dawn of Time, an infinite sequence of Pennies from Heaven fell to Earth
aligning themselves in alternating Halos and Tridents extending over the horizon to the
Twelfth of Never. Raumoko the Maori god of earthquakes plays "Quaboom" with
these coins. He leaps and comes crashing back down to earth so as to cause the coins to
bounce and flip and tumble randomly. The effect of his jumps, however, never reaches
beyond the horizon. The outcome of a game can be thought of as an infinite sequence over
that eventually alternates with at odd indices and at even indices, as below
Are there countably infinitely many or uncountably infinitely many such sequences for
the game "Quaboom"?
Explainprove your answer.
Some sample "games" of Quaboom:
HTHTHTHTHTHTHTHT HTHTHTHTHTHTHTHT
THTHTHTHTHTHTHTH HTHTHTHTHTHTHTHT
HHHHHHHHHHHHHHH HTHTHTHTHTHTHTHT
TTTTTTTTTTTTTTT HTHTHTHTHTHTHTHT
TTTHTHTTTHTHTHTT HTHTHTHTHTHTHTHT
HHT HTHTHTHTHTHTHTHT
b The Divinic Code of a god is made up of a sequence of Supremo Acrids: Genetica,
Theobroma, Cornucopia, and Ambrosia.
Every such infinite sequence of these Supremo Acrids corresponds to a different god.
Are there countably infinitely many gods or uncountably infinitely many?
Explainprove your answer.
Sample Divinic Codes of some of the gods:
Gaia: GGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
Zeus: AGTCAGTCAGTC...TCAGTCAGTCAG...
Odin GTACTAGCACATGATCAGT...GATCAACCAATGACTAG...
Thor: TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
Shiva: ATGCTAGCATGAC...GATTACAGATTACA.
Jobs: ATTGGGCCCCAAAAATTTTTTGGGGGGGCCCCCCCC.
Hint : One of these is countably infinite, the other uncountably infinite.
Hint : One of these is equivalent to the rational numbers the other to the reals
points Bonus
The diagonalization argument FAILS in one of these. Why? points Uncountability Diagonalization
a At the Dawn of Time, an infinite sequence of Pennies from Heaven fell to Earth
aligning themselves in alternating Halos and Tridents extending over the horizon to the
Twelfth of Never. Ruaumoko, the Maori god of earthquakes plays Quaboom with
these coins. He leaps and comes crashing back down to earth so as to cause the coins to
bounce and flip and tumble randomly. The effect of his jumps, however, never reaches
beyond the horizon. The outcome of a game can be thought of as an infinite sequence over
H T that eventually alternates with H at odd indices and T at even indices, as below
Are there countably infinitely many or uncountably infinitely many such sequences for
the game Quaboom
Explainprove your answer.
Some sample games of Quaboom:
HTHTHTHTHTHTHTHT HTHTHTHTHTHTHTHT
THTHTHTHTHTHTHTH HTHTHTHTHTHTHTHT
HHHHHHHHHHHHHHHH HTHTHTHTHTHTHTHT
TTTTTTTTTTTTTTTT HTHTHTHTHTHTHTHT
TTTHTHTTTHTHTHTT HTHTHTHTHTHTHTHT
HHHTTHTHTHTHHTHT HTHTHTHTHTHTHTHT
b The Divinic Code of a god is made up of a sequence of Supremo Acrids: Genetica,
Theobroma, Cornucopia, and Ambrosia.
Every such infinite sequence of these Supremo Acrids corresponds to a different god.
Are there countably infinitely many gods or uncountably infinitely many?
Explainprove your answer.
Sample Divinic Codes of some of the gods:
Gaia: GGGGGGGGGGGGGGG GGGGGGGGGGGGGGG
Zeus: AGTCAGTCAGTC. TCAGTCAGTCAG.
Odin GTACTAGCACATGATCAGT. GATCAACCAATGACTAG.
Thor: TTTTTTTTTTTTTTTT TTTTTTTTTTTTTTTTTT
Shiva: ATGCTAGCATGAC. GATTACAGATTACA.
Jobs: ATTGGGCCCCAAAAATTTTTTGGGGGGGCCCCCCCC.
Hint : One of these is countably infinite, the other uncountably infinite.
Hint : One of these is equivalent to the rational numbers Q the other to the reals R
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