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)