Probability - Set Theory Question

[FrogPad]

So I'm taking a probability class right now. We are going over elementary set theory, and the professor brought something up which seems non-intuitive to me.

He said that a set must have distinct objects, so...

$$A = \{ 1, \,\, 1, \,\, 2, \,\, 3 \}$$

is not properly defined, because the 1 is repeated. Instead, it must be written as:

$$A = \{ 1, \,\, 2, \,\, 3 \}$$

why is this?

I asked him then what do we do with,
$$A = \{ x^2 | x= -1, \,\, 0, \,\, 1 \}$$

and he said that it would be,
$$A = \{ 1, \,\, 0 \}$$

This seems weird, since in a sense we are losing information.

Would someone be so kind to elaborate on why you can't have repeated objects? I want to know why this is.

(Maybe I should have not posted this in the homework section.)

 

[robert Ihnot]

It has to do with the definition of a set, which is a collection of distinct objects.

"A set, unlike a multiset, cannot contain two or more identical elements. All set operations preserve the property that each element in the set is unique. Similarly, the order in which the elements of a set are listed is irrelevant, unlike a sequence or tuple." http://en.wikipedia.org/wiki/Set

 

[FrogPad]

cool, thanks man.

"All set operations preserve the property that each element in the set is unique"

I can live with that :)

 

[Hurkyl]

By definition, the identity of a set is entirely determined by its membership relation.

Your professor is technically wrong on his first point -- in the usual development of set theory {1, 1, 2, 3} is a perfectly good set. It just so happens that

{1, 1, 2, 3} = {1, 2, 3}.​

You can check that the membership relation is identical for both of these: for all x, x is an element of {1, 1, 2, 3} if and only if x is an element of {1, 2, 3}.

 

[FrogPad]

By definition, the identity of a set is entirely determined by its membership relation.

Your professor is technically wrong on his first point -- in the usual development of set theory {1, 1, 2, 3} is a perfectly good set. It just so happens that

{1, 1, 2, 3} = {1, 2, 3}.​

You can check that the membership relation is identical for both of these: for all x, x is an element of {1, 1, 2, 3} if and only if x is an element of {1, 2, 3}.

So, if I define as set as:
A = {1, 1, 2, 3}

This is technically allowed, and is equivalent to {1, 2, 3}

However, if I perform a set operation such as:

A U NULL_SET = B

Set B would then equal, B = {1, 2, 3} correct?

or is it perfectly acceptable to say that after the set operation B = {1, 1, 2, 3}?

In my example above, let's say I want to count the number of times a function is equal to 1. So I define a set,
A = {x^2=f(x)| x = -1, 0, 1}

Am I going about this the wrong way?

 

[matt grime]

Defining

{x^2=f(x)|x=-1,0,1}

doesn't at all help you count the number of times a function is equal to 1. Firstly, x^2=f(x) is an equation (what is f?). I think you meant to write

{x^2 | x=-1,0,1}

But that doesn't helpy ou count the number of times x^2=1 (all we know from what we've written is that there are at least two elemetsn that square to 1, but that had notthing to do with sets at all). You just do it by counting.

 

[FrogPad]

Defining

{x^2=f(x)|x=-1,0,1}

doesn't at all help you count the number of times a function is equal to 1. Firstly, x^2=f(x) is an equation (what is f?). I think you meant to write

{x^2 | x=-1,0,1}

But that doesn't helpy ou count the number of times x^2=1 (all we know from what we've written is that there are at least two elemetsn that square to 1, but that had nothing to do with sets at all). You just do it by counting.

Ok, I think I'm just getting confused on how I am applying set notation. Let me be a little more rigorous in how I am defining things.

Lets say we have a function x^2 with the domain such that x is a subset of the integers (is this ok how I am saying it?), thus:

x = -..., -2, -1, 0, 1, 2, ...

Now if we define a set of all possible values of x^2 we have,

A = {x^2 | x = -..., -2, -1, 0, 1, 2, ...}

Now if we define a subset such that,
B = {x | x is_an_element_of A AND x=1}

What I would want to see is, B = {1, 1} so now I can count the elements B and see that there are in fact two occasions where the function returns the value 1 when I feed it integers. But this is not correct right?

My whole problem seems to be that it seems that we are LOSING information, since B would equal {1} not {1,1} as I wrote above. So counting the elements of the set really does not help me.

Let me point out, that yes this problem of counting is trivial... but I'm trying to understand set notation, and apply it. I am guessing I am applying it wrong.

 

[matt grime]

You want

{x in Z | x^2=1}

this is a two element set {-1,1}

Try to bear in mind: why would the set of square numbers remember what you squared to get there? Sets don't have positions, or multiple elements. If you want something else, then use something else.

 

[FrogPad]

You want

{x in Z | x^2=1}

this is a two element set {-1,1}

Try to bear in mind: why would the set of square numbers remember what you squared to get there? Sets don't have positions, or multiple elements. If you want something else, then use something else.

Ahh yes,
{x in Z | x^2=1}

doing this as such gives me what I want. I was going about it all wrong, as you put "why would the set of square numbers remember what you squared to get there?"

;) thanks man

and yeah, "Sets don't have positions, or multiple elements. If you want something else, then use something else."

If the problem was actually complicated, a C program would have easily allowed me to brute force the answer. I just wanted to try set theory, and with your help I now see the limitations, and benefits. I appreciate the help.