Meaning of function or mapping or transformation in brief words

DEFINITION:

Given two sets X and Y, a transformation (also called a function or mapping) f: X \rightarrow Y of X into Y is a triple (X,Y,G) where G itself is a collection of ordered pairs (x,y), the first element of each pair being an element of X, and the second an element of Y, with the condition that each element of X appears as the first element of exactly one pair of G.

If each element of Y appears as the second element of some pair in G, then the transformation is said to be onto.

If each element of Y which appears at all, appears as the second element of exactly one pair in G, then f is said to be one-to-one. Note that a transformation can be onto without being one-to-one and conversely.

As an aid in understanding the above definition, consider the equation y=x^{2} where x is a real number. We may take X to be the set of all real numbers and then the collection G is the set of pairs of (x,x^{2}). Taking Y to be just the set of nonnegative reals will cause f to be onto. But if Y is all real numbers, or all reals greater than -7, or any other set containing the nonnegative reals as a proper subset, the transformation is not onto. With each new choice of Y, we change the triple and hence the transformation.

Continuing with the same example, we could assume that X is the set of nonnegative reals also. Then the transformation is one-to-one, as is easily seen. Depending upon the the choice of Y, the transformation may or may not be onto, of course. Thus, we see that we have stated explicitly the conditions usually left implicit in defining a function in analysis. The reader will find that the seemingly pedantic distinctions made here are really quite necessary.

If f: X \rightarrow Y is a transformation of X into Y, and x is an element of the set X, then we let f(x) denote the second element of the pair in G whose first element is x. That is, f(x) is the “functional value” in Y of the point x. Similarly, if Z is a subset of X, then f(Z) denotes that subset of Y composed of all points f(z), where z is a point in Z. If y is a point of Y, then by f^{-1}(y) is meant the set of all points in X for which f(x)=y; and if W is a subset of Y, then f^{-1}(W) is the set theoretic union of the sets $f^{-1}(w)$, where w is in W. Note that f^{-1} can be used as a symbol to denote the triple (X,Y,G^{'}) wjere G^{'} consists of all pairs (y,x) that are reversals of pairs in G. But f^{-1} is a transformation only if f is both one-to-one and onto. If A is a subset of X and if f: X \rightarrow Y, then f may be restricted to A to yield a transformation denoted by f|A: A \rightarrow Y, and called the restriction of f to A.

Cheers,

Nalin Pithwa

Fibonacci Numbers: Recreational math

Reference: ICEEM Year 8 Math text book, Australian Mathematical Society.

A Fibonacci sequence is a sequence F_{1}, F_{2}, F_{3}, \ldots of numbers in which each term from the third one onwards is the sum of the two terms that immediately precedes it. You have to have two numbers to start with, F_{1} and F_{2}. These are called seeds. Then,

F_{3}=F_{1}+F_{2}

F_{4}=F_{2}+F_{3}

and so on. The classic Fibonacci sequence has 1 and 1 as its seeds. It first ten terms are:

1,1,2,3,5,8, 13, 21, 34 and 55.

Use a calculator when appropriate in the following:

Activity 1

Write out the classic Fibonacci sequence as far as its 25th term, F_{25}. Before you calculate F_{11}, make a rough guess of what F_{25} will be. See how good your guess turns out to be.

Activity 2

Pick any two numbers as seeds and work out the first 20 terms for that Fibonacci sequence. Pick entirely different seed numbers from the person besides you (example, your friend, or teacher, or any one else imaginary :-)), and keep your list reasonably neat, as we will be coming back to it in a little while.

Activity 3

Swap your two seed numbers from Activity 2 around and figure out the first twenty terms in the new Fibonacci sequence. (If, your sequence in Activity 2 started 6, 11, 17, 28, 45, …) your new sequence will start 11, 6, 17, 23, 40, ….). Yes, you do get quite different numbers from Activity 2.

Activity 4:

It is now time to make a few observations about your Fibonacci sequences.

  • The classic sequence in Activity 1 has two odd seeds. This gives a certain pattern of odd and even terms through out the sequence. What happens if you start with two even seeds, or an odd and even seed? Explain.
  • Compare the 10th terms you generated in each of the sequences in Activity 2 and Activity 3. Which one is larger? Compare the 20th term as well. Can you explain what is happening?
  • Use the calculator to divide the term F_{10} by the term F_{9} immediately before it. Write your answer down. Then, do the same with the second sequence. Now repeat the calculation with terms F_{20} and F_{19}. Do you notice anything interesting? Did any of the “other people” who are doing this activity get the same ratio? They should all have got the same ratio. They should all get the same number, even though there may be v small diferences in the sixth decimal places.
  • For the classic Fibonacci sequence, the first two terms larger than 1000000 are F_{32}=1346269 and F_{33}=2178309. Use these two values to see if what you noticed in the previous ratio calculations still holds for higher order terms in the classic Fibonacci sequence.

The number you obtained (to a good approximation) in the ratio calculations is famous and interesting enough to deserve its own Greek letter. It is called \phi (pronounced to rhyme with spy). It is called the golden ratio or golden mean. It is a very interesting number with long history.

Search Google and you will discover some amazing facts about \phi. It appears in many different ways in geometry and architecture.

Now try calculating these values and see what you notice about them:

a. \phi^{2}

b. \frac{1}{\phi}

c. (2\phi -1)^{2}

Regards,

Nalin Pithwa.