How to define a function involving strings?

[DevNeil]

Homework Statement

We have a problem set on my Discrete Mathematics class:

Let X be the set of strings over {a,b} of length 4 and let Y be the set of the strings over {a,b} of length 3. Define function f from X to Y by the rule:

f(alpha) = string consisting of the first three characters of alpha.

Is f one-to-one? is f onto?

The Attempt at a Solution

I don't know where to start but what I understand on the problem is that X is a set with {aaaa,bbbb,aaab...etc} and Y has {aaa,bbb,aab...etc}. Do I need to list all the elements of the sets? I was also confused by the alpha variable. Hoping for your answers. Thanks.

 

[clamtrox]

Alpha is any element in the set X, so any string of length 4. Your function f takes an element of X and maps it into an element of Y. For example, f(bbaa) = bba.

What does it mean for f to be one-to-one or onto?

 

[DevNeil]

Alpha is any element in the set X, so any string of length 4. Your function f takes an element of X and maps it into an element of Y. For example, f(bbaa) = bba.

What does it mean for f to be one-to-one or onto?

You need to state if it's injective(1to1) or surjective(onto) or bijective(injective && surjective). I can say that this is not one-to-one (injective) because as counterexample f(aabb) = aab , f(aaba)= aab, same result but different alpha.

Can you also tell me how to define the function formally?

 

[clamtrox]

You need to state if it's injective(1to1) or surjective(onto) or bijective(injective && surjective). I can say that this is not one-to-one (injective) because as counterexample f(aabb) = aab , f(aaba)= aab, same result but different alpha.

And then you only need to figure out if it's onto. That should be straightforward too.

Can you also tell me how to define the function formally?

The definition you were given looks pretty formal to me. Is there something wrong with it?

 

[DevNeil]

And then you only need to figure out if it's onto. That should be straightforward too.


The definition you were given looks pretty formal to me. Is there something wrong with it?

Ohh... I am thinking of a function that was mathematically defined. So another question, is this an acceptable definition of a function? I am having a hard time defining it. Thanks for your time :)

 

[clamtrox]

Ohh... I am thinking of a function that was mathematically defined. So another question, is this an acceptable definition of a function? I am having a hard time defining it. Thanks for your time :)

A function f:X->Y is just a rule that gives you a unique element of Y for each element of X. In this case, the rule is simple: drop the last digit.

 

[vela]

You could just list all 16 ordered pairs (x,f(x)) in X×Y.