For the first week, we explore basics of functions and permutations, picking up basic notation that we will use throughout the course.

On Tuesday, we obtained a definition of a function after first learning an informal definition of 'set' and examining a number of examples of functions.  Along the way, we picked up some basic set notation, as well.  We then learned definitions of surjective (onto) functions and injective (one-to-one) functions, while getting a bit of practice in manipulating the notation, and figuring out if a given function is injective or surjective.  We also took a bit of time to look at polynomials on the finite field Z5.

Associated Reading: The Art and Craft of Problem Solving, Section 5.1 (pg 143), and the Wikpedia articles  on sets and these kinds of functions.

On Wednesday, we continued discussion of injective functions, learned about bijections, and the cardinality of different kinds of sets.  We proved a theorem stating that if a bijection exists between two sets, then the sets have the same cardinality.  Additionally, we found that if a function f:S->T has an inverse function, then f must be a bijection.  We began to wonder about the cardinality of infinite sets, which we will explore further soon!

Associated Reading: As above, see the Art and Craft of Problem Solving and Wikipedia (especially on bijections).

Homework, Week One

1) Let A and B respectively denote the even and odd integers (recalling that 0 is even).  Is there a bijection from A to B?  From B to A? Is there a bijection from Z to A?

2) Find a bijection between the set of whole numbers N and the even whole numbers 2N = { 2i | i in N }.

2) (Challenge) (AIME1984) The function f:Z->Z satisfies f(n)=n-3 if n>=1000, and f(n)=f(f(n+5)) if n<1000.  Find f(84).

3)  (Challenge) (AIME1984) A function f:R->R satisfies f(x-2)=f(x+2) and f(x-7)=f(x+7) for all x.  If x=0 is a root of f (ie, f(0)=0), what is the least number of roots that f must have in the interval -1000<=x<=1000?

### 6 Responses to SMA109: Week One

1. HARRISON LUMUMBA says:

wonderful but though the introductory bit seems to be hard,we hope to get used to it and understand but bravo all in all.

2. jecinta mulongo says:

• tom denton says:

Hi, Jecinta;
We're going to print out copies of a couple books and leave them in a central location. We'll announce in class when it's done.
Best,
-tom

• jecinta mulongo says:

Thanx

3. jeffar junior says:

Hi, I have this problem, I joined mathematical science late this semester because I applied for interfaculty transfer and Its like I'm having difficulties in understanding the SMA 109 concepts. I really like this coarse and I have tried by all means to reach that understanding level through group work but I still cant get along, please help. my number:0713182792

• tom denton says:

Hi, Jeffar;

The best thing to do would be to find a group of colleagues to help you understand what we've done so far. We'll then be able to answer any questions you have that your colleagues can't help you with. (We're very happy to talk with you and help, but it will be too much time to reproduce the whole course!) Feel freet o approach us after class, as well, or in the maths lab, where we can often be found working.

Best,
-tom