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:

1. Section 0, problems 12 and 16.
2. Section 1, problems 3 and 13.
3. Section 2, problems 2, 3, 7, 10, and 11.

--------------------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)