Math 360, Fall 2013, Assignment 2

By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so worthy of my consideration.

- Mary Shelley, Frankenstein

Carefully define the following terms, then give one example and one non-example of each:

1. Function (from a set \(A\) to a set \(B\)).
2. Injection.
3. Surjection.
4. Bijection.
5. Same cardinality.
6. Root of unity.
7. Binary operation.
8. Closed (under a given binary operation).
9. Associative.
10. Commutative.
11. Composition (of two functions).

Carefully state the following theorems (you need not prove them):

1. Theorem relating equivalence relations to partitions.
2. Associativity of composition.

Solve the following problems:

I.) Section 0, problems 12 and 16.

12.) Let \(A=\{1,2,3\}\) and \(B=\{2,4,6\}\). For each relation between A and B, given as a subset of their cartesian product, decide whether it is a function that maps A to B. If it is, determine whether it is an injection and whether it is a surjection.

a.) \(\{(1,4),(2,4),(3,6)\}\)

b.) \(\{(1,4),(2,6),(3,4)\}\)

c.) \(\{(1,6),(1,2),(1,4)\}\)

d.) \(\{(2,2),(1,6),(3,4)\}\)

e.) \(\{(1,6),(2,6),(3,6)\}\)

f.) \(\{(1,2),(2,6),(2,4)\}\)

16.) List the elements of the power set of the given set, and give the cardinality of the power set (Hint: There is a theorem that relates the cardinality of a set A and its power set P(A))

a.) \(\emptyset\)

b.) \(\{a\}\)

c.) \(\{a,b\}\)

d.) \(\{a,b,c\}\)

II.) Section 1, problems 3 and 13.

3.) Compute the following arithmetic expression and give the answer in the form of \(a+bi\) for \(a,b \in \mathbb{R}\)

\[i^{23}\]

13.) Write the given complex number in the polar form.

\[-1+i\]

III.) Section 2, problems 2, 3, 7, 10, and 11.

2.) Compute (a * b) * c and a * (b * c). Can you say on the basis of these computations whether * is associative?

3.) Compute (b * d) * c and b * (d * c). Can you say on the basis of these computations whether * is associative?

For the following 3 exercises, determine whether the binary operation * defined is commutative and whether * is associative.

7.) * defined on \(\mathbb{Z}\) by letting \(a * b = a - b\)

10.) * defined on \(\mathbb{Z}_{> 0}\) by letting \(a * b = 2^{ab}\)

11.) * defined on \(\mathbb{Z}_{> 0}\) by letting \(a * b = a^b\)

--------------------End of assignment--------------------

Questions:

1) I still don't get what non - example means for math. Can someone please explain?? Thank you!!

I believe what a non-example would be is something that, for a reason you identify, is not what has been defined. For instance, we know that a function can take an input to at most one output, so perhaps you could say that a function from the real numbers to the real numbers that contains the ordered pairs (1,3) and (1,5) is not a function because it assigns two different outputs to the same input (in that case, we would say it is a relation, but not a function). I hope this helps. --Robert.Moray (talk) 16:29, 11 September 2013 (EDT)

Yes, that's what I mean. The idea is to build intuition for what a word means by thinking of some objects that fit its definition, then thinking of some objects that don't. (I'm only asking you to write down one of each in the assignment, but you will do better in the course -- and enjoy the mathematics more -- if you privately think about as many examples and non-examples as you can.) Of course it's always easy to come up with extreme non-examples -- e.g. "a blade of grass is not a function" -- but it's more helpful to think about near misses like the one that Robert gave above, since they shine more light on the boundary of the word's meaning. --Steven.Jackson (talk) 08:18, 12 September 2013 (EDT)

2.) FYI: This is my attempt at proving \(\mathbb{R}\) is uncountable. Anyone who saw Professor Wortman do this in MATH 280 is probably familiar with this proof. Feel free to offer suggestions! I used LaTeX that was particularly difficult to replicate on MediaWiki so that is why it's a screenshot instead of directly on the wiki (and also to save space on the page). --Robert.Moray (talk) 12:09, 12 September 2013 (EDT)