DQ’s
1. Leopold Kronecker famously (infamously?) wrote: “God made the integers; all else is the
work of man.”
Later, an almost famous mathematician stated “Nothing made the integers”. This is not at
all a declaration or statement of faith, but rather a marvel at the wonder of set theory.
Thinking about sizes of sets answer the following:
1. What are the orders of the following sets?
a. 
b. {}
c. {,{}}
2. Take the numerical value of the month you were born (I.e. January = 1, February =
2,…) and add 3 to it. Call this value B. Write out the set composed of empty sets with
an order of B.
3. In general, we see that the Natural Numbers can be built from the empty-set. One
might even be compelled to say that “Nothing (the empty set) made the Natural
Numbers.” Can the Integers, in fact, be made from nothing? If so, how?
2. What does it mean in mathematics when we say that a statement is “vacuously true”?
How is this different from saying that a statement is trivially true, or that a claim is
trivial? Give an example of a vacuously true statement, and give an example of
statement that has a trivial case.
3. Explain the logic behind a proof by contradiction. That is, suppose we are attempting to
prove PQ by way of contradiction. How do we set up this implication for a
contradiction? Make a truth table that demonstrates why the contradiction must prove
the implication is true.