Properties of Relations on X (Equivalence relation)

[cdnrice]

First question:
Suppose X' is a partition of X. Define the relation:
R = {(x, y) ∈ X × X : ∃Y ∈ X' , x ∈ Y ∩ y ∈ Y }

Show that R is reflexive, symmetric and transitive.

Second question:

If f : X → Y and g : Y → X are functions, then g ◦ f is a function.

Would anyone care to solve these two questions for me? I've been working on them for hours, more so the latter but the first question I cannot grasp my head around it.

 

[stapel]

First question:
Suppose X' is a partition of X. Define the relation:
R = {(x, y) ∈ X × X : ∃Y ∈ X' , x ∈ Y ∩ y ∈ Y }

Show that R is reflexive, symmetric and transitive.

Second question:

If f : X → Y and g : Y → X are functions, then g ◦ f is a function.

Would anyone care to solve these two questions for me? I've been working on them for hours, more so the latter but the first question I cannot grasp my head around it.

Since you've "been working on them for hours", you'll have plenty of work you can show us. Please reply with that info, so we can see where you're having difficulties. Thank you!

 

[pka]

First question:
Suppose X' is a partition of X. Define the relation:
R = {(x, y) ∈ X × X : ∃Y ∈ X' , x ∈ Y ∩ y ∈ Y }
Show that R is reflexive, symmetric and transitive.

I agree that you need to post your work. But I think you are really confused about notation.

What does it mean to say that \(\displaystyle X'\) is a partition of \(\displaystyle X~?\)

You should write the relation as:
\(\displaystyle R = \left\{ {(x,y) \in X \times X:\exists Y \in X'\;\& \;\{ x,y\} \subset Y} \right\}\)

Now if you understand partitions then if \(\displaystyle t\in X\) then \(\displaystyle \exists T \in X'\) such that \(\displaystyle \{t,t\}\subset T\). So \(\displaystyle R\) is reflexive. WHY?

Second question:
If f : X → Y and g : Y → X are functions, then g ◦ f is a function.

The second is not true in general. The function \(\displaystyle g\) must be surjective (onto) for it to be true.

Now show us your work.

 

[cdnrice]

Hi sorry for the late reply.

For the 2nd question I thought the statement was true.

Let X=(1,2,3,4,5,6)
Let Y=(a,b,c,d,e,f)

f=[(1,a),(3,c),(5,e)]
g=[(a,2).(c,4),(e,6)]

then
gof=[(1,2),(3,4),(5,6)].

Can you explain why it has to be surjective to be true.

As for the 1st question, I have no idea where to start :S So any help would be greatly appreciated. Once again sorry for the late reply back.

 

[cdnrice]

R = {(x, y) ∈ X × X : ∃Y ∈ X' , x ∈ Y ∩ y ∈ Y }

The original relation in question 1 was wrong, posted above is the correct relation by the way.

 

[pka]

Hi sorry for the late reply.

For the 2nd question I thought the statement was true.

Let X=(1,2,3,4,5,6)
Let Y=(a,b,c,d,e,f)

f=[(1,a),(3,c),(5,e)]
g=[(a,2).(c,4),(e,6)]

Neither of those is a function. Do you know why?

 

[Ishuda]

Neither of those is a function. Do you know why?

I would say not only are they a function but they are a one to one function mapping a subset of X onto a subset of Y for f and mapping a subset of Y onto a subset of X for g, see
http://www.purplemath.com/modules/fcns.htm
for example

 

[pka]

I would say not only are they a function but they are a one to one function mapping a subset of X onto a subset of Y for f and mapping a subset of Y onto a subset of X for g, see
http://www.purplemath.com/modules/fcns.htm
for example

Then you don't know the set theoretical definition of a function.
A function is a set of pairs such each term of the initial set is the first term of some pair in the function.

 

[Ishuda]

Then you don't know the set theoretical definition of a function.
A function is a set of pairs such each term of the initial set is the first term of some pair in the function.

Sort of a restricted definition isn't it? How about functions in 3 (triplets) or more dimensions? If you are not going to allow subsets, how about let R be the set of real numbers and consider the relationship \(\displaystyle (x, \frac{1}{x-2}), x \epsilon \text{{R - {2}}}\). Isn't that a function?

As for the example, let us take a proper subset of X, say X' = {1, 3, 5}. The function f maps X' into the set Y and onto the set Y'={a, c, e}. So, by your definition X' would be the initial set or, from the link you give, X' would be the domain and Y (or Y') ithe co-domain. A similar situation applies for the function g.

 

[pka]

Sort of a restricted definition isn't it? How about functions in 3 (triplets) or more dimensions? If you are not going to allow subsets, how about let R be the set of real numbers and consider the relationship \(\displaystyle (x, \frac{1}{x-2}), x \epsilon \text{{R - {2}}}\). Isn't that a function?

Well of course it is a function from the set \(\displaystyle \mathbb{R}\setminus \{2\}\) to \(\displaystyle \mathbb{R}\).

Functions are not about dimensions. They occur for all dimensions, and the definition is the same.

Now the OP is about finite sets. In that case the initial is the set of ordered pairs