What is one to one correspondence?

Answer 1

A one to one correspondence has to do with mathematical functions. But to understand this we have to understand how a function is made, and since you ask your question I guess there is a slight lack of understanding concidering the definition of a function. I could just give you a definition of "1-1 correspondence", but that won't help if you don't understand what a function really is. So I hope you'll take your time and read through this a couple of times, making sure you understand all the bits and pieces.

Let's look at two mathematical sets A and B. A set is just a collection of elements. So we let A = {1,2,3,4}, and B = {k,g,z,a}. The elements in a set can be whatever we want them to be. Bananas, televisions, numbers or letters. Here we have chosen numbers for one and letters for the other. So, what do sets have to do with functions? Well, everything. It's what we use as a basis to create and define the functions we want to work with.

So, let's try and make a function f(x) from Ato B. Saying "from A to B" means that all the values we put into f(x) must come from the set A, and all the values coming out of the function must come from the set B. So the only things we are allowed to feed our function are the numbers 1, 2, 3 and 4, and the only thing that can come out of the function has to be either k, g, z or a. The "from A to B"-part is usually skipped in mathematical textbooks, but all functions are defined from one set to another. Tt's a very essential part of functions, and to understand the concept of one to one correspondence it is crucial to understand this bit.

Since we're the boss of our function f(x), we can decide what the output of the function should be. So this is what we decide: f(1) = g, f(2) = k, f(3) = a and f(4) = z. Now you see that each input value will have it's own output value. Each number from A will correspond to some letter in B. Neat and tidy. But there's nothing wrong in changing our mind, saying that both f(1) andf(2) will be g. We see that k has never been used as an output, but that's perfectly fine. All the elements of our output set don't have to be used. If we use our first desicion of values, the function has a one to one correspondence between Aand B. Our second choice of output values destroys the one to one correspondence.

So a formal definition could be said to be something like this: "A function is said to have a one to one correspondence if no two input values give the same output value"

Some concequences of this definition:

- The set B can never be smaller than the set A if the function is 1-1 (one to one)

- Graphically, a 1-1 function has to be strictly increasing or decreasing in value. It can never swing up and down

If you have understood this, you should be able to answer these questions without too much trouble. Take your time, draw the functions and reflect on the theory above (answers found below):

1) Is y(x) = x a 1-1 function?

2) How about f(x) = sin(x)?

3) Why can't the set B be smaller than A, if the function is supposed be 1-1?

4) Which two sets is the function y(x) = x usually defined from/to? "y(x) = x, from ? to ?.

5)

a)

If we define f(x) = cos(x) and let A (the input set) be all the numbers from 0 to Pi, how small can we let the output set be, and still cover all the input values?

b)

Is this function 1-1? Remember that we define it only up to the value cos(Pi).

c)

Will there be any difference to answers in a) and b) if we let A be the set of all numbers from 0 to 4*Pi?

Answers:

1) and 4):

The function y(x) = x is usually defined from R to R, where R is the set of all real numbers. (This is the way most textbook functions are defined) Since y(x) is always increasing in value, it will be a 1-1 function.

2)

Both sin(Pi) and sin(3*Pi) have the value 0, so it contradicts the definition of a function being 1-1.

3)

Let's assume the number of elements in B is less than the number of elements in A. A function has to have a defined output value for all of the input values. So ultimately, as we go through the list of input values, we will run out of elements in the output set. This means that if our starting assumption is true, least two inputs of f will have the same output value.

5)

a)

We know cos(0) = 1 and cos(Pi) = -1, and that cos(x) generally swings between 1 and -1. So here the set B=[-1,1] will be enough.

b)

Yes, the function is always decreasing from 0 to Pi.

c)

for a) the answer would be the same, but for b) we would have to change our answer. After cos(Pi), the function starts increasing again, so we will get output values that have already occured.