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Hello Off-Topic.
I am here to discuss mathematics with you again....
We will discuss set-builder notation below:
Sets are merely mathematical "structures"/objects that enclose other mathematical objects.
Typical set notation is through the use of the curly-bracket characters, {}.
Sets are quite useful for modeling the properties of "concrete", real-world objects.
A sufficient example would be a young child's playpen. The playpen can contain many "toy" objects. Those "toy" objects are enclosed within the playpen. Hence, they are "elements" or "members" of the playpen.
Mathematically, we could express the playpen(And its elements) as this:
P = {Rubber Car,Brunette-Doll,Dinosaur};
"P" refers to the actual playpen set. All of the remaining names merely refer to the members of the playpen.
There is an inherent issue with the above: It assumes that all elements within the set are always toys. Well, since it is preferable to have only "child-safe" objects within a playpen, it is necessary to ensure that only permissible objects will be added into our set. The below example accomplishes this:
P = {o | T(o) = 1}
To a novice, the above may seem rather "complicated", though I assure you that the concept is actually much simpler than it appears.
"o" is the actual object that may become a member of playpen P. Function "T" accepts a object as its input and determines whether the input object is an actual toy.
If the object is a toy, 1 is the output value. If the object is not a toy, 0 is the output value.
Consequently, in this example, all objects that are toys(And consequently output 1) are enclosed within the playpen. Therefore, they are members of set P.
Mathematically, notation such as "x | condition" means in regular language that "all x values that fulfill a specific condition truthfully are members of a given set"
As you can see from our previous examples, functions and sets seem to synchronize perfectly together when used appropriately.
Hopefully this has provided an interesting subject of discussion for many users. |
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ZecritJoin Date: 2013-01-24
Post Count: 2618 |
But 2 + 2 = 5. |
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Well then, does anyone else on the Off-Topic subsection also enjoy set notation, other than Zecrit? |
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Let us use another basic example.
Suppose that I have a box of crackers:
B = {Cracker One, Cracker Two, Cracker Three, ...}
Within this particular scenario, set B has an infinite quantity of elements. Thus, there is a boundless quantity of crackers within the box.
Now then, I need to determine whether the appropriate "cracker" objects are within the box.
Doing so will require some rather specific details.
Firstly, I can create function "S" to handle both operations simultaneously. Function "S" will check the state of the cracker and determine whether it has valid saltine content and round characteristics. If so, it will output "s" for "saltine" and "r" for "round"; otherwise, it will output "n" for "no."
Mathematically, this is how our function "S" would appear:
B = {c |s ∈ R(S(c)) ^ r ∈ R(S(c)))}
The example provided above may be somewhat harder to understand. Nonetheless, I will explain it in a basic manner.
"B" is the original set of crackers. "c" is the potential cracker object we need to analyze.
The special "e" symbol can be translated as "Object o is a member of set S." Here is an example:
a ∈ A
Within the above example, "a" is a letter within the English alphabet. Consequently, "a" is a member of a set "A", which refers to the alphabet.
Mathematical functions have two sets within themselves: Domain and Range. The "Domain" represents the input values. The "Range" represents the output values.
Since the "Range" represents the output values, the returned cracker property values are stored within the Range.
"R(S(c)" will retrieve the Range of function S as it outputs input "c" values.
"^" is intended to represent the logical AND operator. It ensures that both conditions are true.
Inevitably, if "n" and "s" are genuine elements of the Range set of function S, the object is considered to be a cracker. In all other conditions, the object is not a cracker.
This is yet another example of set manipulation. |
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Alright. The thread has been revived for positive purposes.
Feedback is welcomed. |
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Well then, I suppose another instance of revival is necessary.
I will compose another entertaining sample for my fellow users soon enough. |
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Actually, I have decided to postpone any further samples until I have received more feedback.
Surely another user would like to join our discussion, yes? |
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AetricityJoin Date: 2012-03-29
Post Count: 51079 |
Since I am so used to programming, at first I read the "|" as the OR bitwise operator, which made this thread a bit nonsensical at first. This seems like a very simple if else with classes type condition to check whether or not x is in class y. I have no idea how this would spark a subject of discussion; enlighten me. |
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"I have no idea how this would spark a subject of discussion"
From a mathematically-pure perspective, it is intended to discuss the potential uses of set-builder notation as a modeling tool.
Nonetheless, It is fairly interesting that you easily visualized a conditional statement algorithm.
After-all, theory precedes application. |
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AetricityJoin Date: 2012-03-29
Post Count: 51079 |
"From a mathematically-pure perspective, it is intended to discuss the potential uses of set-builder notation as a modeling tool.
Nonetheless, It is fairly interesting that you easily visualized a conditional statement algorithm.
After-all, theory precedes application."
It is so; usually I visualize everything in the most logical processes possible: code. In many ways it is helpful to perceive of everything as mere chains of events caused by the randomness unmatchable by any pseudo-random generator.
Speaking of application, though, may I ask, are you working on any application at the moment? If so, give me a brief explanation of it, I'm pretty interested to know of any new projects coming about. |
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"may I ask, are you working on any application at the moment"
Unfortunately, I am not developing any projects at the current moment, due to organizational purposes.
Have you begun development of any "intriguing" projects, other than your AI? |
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AetricityJoin Date: 2012-03-29
Post Count: 51079 |
"Have you begun development of any "intriguing" projects, other than your AI?"
I have begun working on a "scripting language," I guess you could call it, in C++. I plan this on being a very simple project since I don't plan on making it too complex. You'd program to it by making an array of commands. I plan on making the syntax and keywords extremely simple, and in a way, like assembly. You can only do one command per line, although I might actually make it so you can split things using an ampersand for the hell of creativity. In a way, I want it to be user-friendly, but this is really only going to be a project for myself since it's not going to be released since I do not have the proficiency to embark on something great enough for distribution, although if I'm satisfied with the outcome, I might remake it and actually do something with it.
--but I'm getting ahead of myself.
If you want to know, this is how the commands will be set up like:
"out 0 hello 1"
What that does is it adds a string (hence 0) with the second parameter being hello to the output stream 1. I plan on adding 16 output streams so then you can store 16 different outputs and you could merge them and stuff; it's a bit weird and all, but I really want to just make a bit of an abstract language.
What you can then do with it is print it out, so:
"pout 1", where pout is the command for printing out to output and "1" is the output stream.
Boolean logic and that stuff works like in pretty much all programming languages I know, except for the sake of creativity, I decided to mix it up a bit.
"statement [condition] run [ -=- code here ] notmet [ -=- code here ]"
-=- is used for commenting.
essentially instead of if (condition) { or if condition then it's statement [x > 5] run ( )
Really, this language is not being made for any practical purpose as I have previously stated, but it'd be cool to work on for fun and maybe write some programs with.
I only have the very beginning done, though. |
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AetricityJoin Date: 2012-03-29
Post Count: 51079 |
I was joking about it being simple by the way, I want to make it a little bit difficult for the hell of it.
I mean to your every-day programmer I'm assuming something like: "statement [x > 5] run" would be a bit weird, but hey, it's something new I guess.
Still though, it's going to only incorporate very basic functionality |
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