Mathematical Thinking
Due November 28
3:2; 1-8, 13-19, 28
{3, 6, 12, 15, 18, ….} It’s the set of multiples of 3 except that the multiple 9 is missing. This may or may not be intented.
{1, 2, 3, 4, 5} This is the five element set consisting of the first five counting numbers.
{1/2, 1/4, 1/8, 1/16, ….} This is the set of positive integer powers of ½.
{-1, -2, -3, -4, ….} This is the set of negatives of counting numbers.
{1, 4, 9, 16, 25, 36, 49, 64, 81, 100} This is the set of squares of the first ten positive integers.
The second and fourth are infinite. The first and third are finite.
This is a hard idea. For finite sets, it means the number of elements. For infinite sets, it means roughly speaking, the types of sets which can be put into a one-to-one correspondence with. Those that can be put in one-to-one correspondence with N are called countable. Two sets that have the same cardinality are sometimes called equinumerous. In class we proved that the set of real numbers has a greater cardinality than N.
(so E = {2,4,6,8,….}) and O stand for the set of all odd natural numbers
(so O = {1,3,5,7,….}). Show that the sets E and O have the same cardinality by describing an explicit one-o-one correspondence between the two sets.
The mapping that sends each number to the one that is one bigger is one-to-one from the set O to the set E.
(so E = {2,4,6,8,….}). Show that the set E and the set of all natural numbers have the same cardinality by describing an explicit one-to-one correspondence between the two sets.
The mapping that sends each number to its double is one-to-one from the set N to the set E.
for “ends in five”). That is, EIF = {5, 15, 25, 35, 45, 55, 65, 75,….}. Describe a
one-to-one correspondence between the set of natural numbers and the set of EIF.
The mapping that sends each number n to 10n-5 is one-to-one from the set N to the set EIF.
13. Squaring off. Let S stand for the set of all natural numbers that are perfect squares, so S = {1, 4, 9, 16, 25, 36, 49, 64, ….}. Show that the set S and the set of all natural numbers have the same cardinality by describing an explicit one-to-one correspondence between the two sets.
The mapping that sends each number n to n^2 is one-to-one from the set N to the set S.
14. Counting cubes (formerly crows). Let C stand for the set of all natural numbers that are perfect cubes, so C = {1, 8, 27, 64, 125, 216, 343, 512, ….}. Show that the set C and the set of all natural numbers have the same cardinality by describing an explicit one-to-one correspondence between the two sets.
The mapping that sends each number n to n^3 is one-to-one from the set N to the set C.
15. Reciprocals. Suppose R is the set defined by
R = {1/1, 1/2, 1/3, 1/4,1/5, 1/6, ….}
Describe the set R in words. Show that it has the same cardinality as the set of natural numbers.
The mapping that sends each number n to its reciprocal 1/n is one-to-one from the set N to the set R.
The manager simply sends each guest to the room that has a number one bigger.
The manager simply sends each guest to the room that has a number two bigger, leaving rooms 1 and 2 vacant for the new arrivals.
The manager simply sends each guest to the room that has a number twice as big as his original room. This leave all the odd numbered rooms for the new arrivals.
I do not know how to do this without begging the question. My method originally was to assume the universe had finite weight, and that the average grain of sand has a positive weight, so the maximum number of grains is the fraction (weight of universe/average weight of a grain), however, this assume a lot about our universe.
If both barrels ended up finite, say with m and n ball respectively, then the original set had only m+n balls, a contradiction.