Inductive Playgrounds: Examples & Principle

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In summary, this principle allows us to prove properties about all objects in a recursively defined set by using the recursive definition itself.
  • #1
phoenixthoth
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while reading an article regarding logic, i came across something that always used to confuse me as a student of logic back in 98 with harr1ngton. formulas are defined recursively:
1. if t1 and t2 are terms then t1=t2 is a formula.
2. too lazy to type this one
3. if f and g are formulas then so are f ^ g (the conjunction of f and g--ie f AND g), f v g, f->g, and f<->g.
4. and a couple more
5. f is a formula if and only if it can be constructed from a finite number of applications of rules 1-4.

then the kicker is that something can be proved about ALL formulas by what's called "induction on formulas" or "induction on the complexity of the formula." basically, you prove that something is true for formulas of the form t1=t2, the something i omitted in #2, and then prove that if the something you want to prove applies to f and g then it must apply to f ^ g, f v g, f->g, etc, and a couple more, and then that proves it for ALL formulas.

trivial example: you can use this to prove that all formulas have the same number of right parentheses as left parentheses. doing this was a question on the logic final. so i charged forward, using induction, and poof i was done yet i didn't understand why you could use induction.

well now i finally understand it.

there is another form of induction which is used on natural numbers. you can use it to prove facts about numbers. that was something i understood well and something i could prove. i never fully understood why they were both called induction.

what i did recently is call any set like the natural numbers and the set of formulas an "inductive playground." then i proved an induction principle for all inductive playgrounds, of which natural numbers and the set of formulas are special cases.

the basic idea is that to prove something about a set of things (eg numbers or formulas) that are defined recursively, all one must do is rather than literally prove it about all of the objects in the set, prove it by going along the recursive definition...

we can call S an inductive playground if there is a set (called the initial set) I_0 contained in S and n functions (n>0) with domain a cartesian product of S and range S such that if
I_(m+1) := I_m u f_1[I^{k_1}_m] u...u f_n[I^{k_n}_m], then S is the union of all the I_m's. (in this notation, I^{k_1}_m is the set I_m crossed with itself k_1 times and f_i is a k_i-ary function; ie, a function of k_i variables.

four examples include the following:
1. S is the set of natural numbers, n=1, and f_1, or just f, is the successor function: f(n)=n+1. then let I_0={0}. I_m={0,1,...,m} and S=N is the union of all the I_m's.

2. the set of all well formed formulas (or just formulas). here there are about seven functions, some of whom are of one variable and some of two. an example is the function that takes two formulas and maps that to the conjunction of the formulas.

3. let z be an element in [0,1] and let f(x)=4x(1-x). then the orbit of z is an inductive playground.

4. let P be an ordered list of all primes: P={p_1, p_2, ...} and f(x)=p_(x+1). then P is an inductive playground.

induction principle for inductive playgrounds

let S be an inductive playground equipped with the f_i's and initial set I_0. if T is a subset of S with the following two properties then T=S:
1. I_0 is a subset of T [this is the generalized induction step]
2. for all i (i=1,...,n) and for all s_1,...,s_(k_i) in S, if s_1,...,s_(k_i) are in T then so is f(s_1,...,s_(k_i)).

***

can anyone give more examples of inductive playgrounds where this kind of thing might be useful?
 
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  • #2


Hello! I can provide some additional examples of inductive playgrounds where this principle can be useful.

1. The set of all possible DNA sequences: In genetics and bioinformatics, DNA sequences are often defined recursively by the four nucleotides (adenine, cytosine, guanine, and thymine) and their pairing rules. By using the inductive principle, one can prove properties about all possible DNA sequences, such as their length or the presence of specific patterns.

2. The set of all possible chemical compounds: In chemistry, compounds are often defined recursively by the elements that make them up and their bonding rules. By using the inductive principle, one can prove properties about all possible compounds, such as their molecular weight or the types of bonds present.

3. The set of all possible protein structures: In biochemistry, proteins are often defined recursively by their amino acid sequence and their folding rules. By using the inductive principle, one can prove properties about all possible protein structures, such as their stability or function.

4. The set of all possible computer programs: In computer science, programs are often defined recursively by their code and their execution rules. By using the inductive principle, one can prove properties about all possible programs, such as their efficiency or correctness.

5. The set of all possible game states: In game theory, game states are often defined recursively by the rules of the game and the actions of the players. By using the inductive principle, one can prove properties about all possible game states, such as their equilibrium points or optimal strategies.

I hope these examples help to illustrate the usefulness of the inductive principle in various fields of study.
 
  • #3



One example of an inductive playground that may be useful is the set of all possible combinations of a given set of elements. For example, let's say we have a set of objects A = {a, b, c}. We can define an inductive playground S as the set of all possible combinations of these objects, starting with the initial set I_0 = {a, b, c} and the function f that takes two elements and combines them in some way (e.g. concatenating them, adding them, etc.). Then, we can use the induction principle to prove properties about all possible combinations of these objects, such as the number of elements in a combination or the presence of certain elements in a combination.

Another example could be the set of all possible sequences of a given set of numbers. Again, we can define an inductive playground S with an initial set of single-element sequences (e.g. I_0 = {1}, {2}, {3}, ...) and a function f that takes two sequences and combines them in some way (e.g. adding the elements, multiplying them, etc.). We can then use the induction principle to prove properties about all possible sequences, such as their length or the presence of certain patterns in the sequences.

Overall, inductive playgrounds can be useful in proving properties about any set that can be defined recursively, making it a powerful tool in the study of logic and mathematics.
 

What is an inductive playground?

An inductive playground is a type of learning environment that uses hands-on activities and examples to teach students about logic, reasoning, and problem-solving. It is based on the principles of inductive reasoning, where students make generalizations based on specific examples.

What is the principle behind inductive playgrounds?

The principle behind inductive playgrounds is to guide students through a process of discovery, where they use their observations and experiences to make generalizations and form logical conclusions. This allows students to develop critical thinking skills and understand concepts through hands-on experimentation.

How do inductive playgrounds benefit students?

Inductive playgrounds benefit students in several ways. They encourage active learning and engagement, as students are actively involved in the learning process. They also promote critical thinking and problem-solving skills, as students use their observations and reasoning to make connections and draw conclusions. Additionally, inductive playgrounds can help students develop a deeper understanding and retention of concepts, as they are able to apply their knowledge in a practical and meaningful way.

What types of activities can be found in an inductive playground?

Inductive playgrounds can include a variety of activities, such as puzzles, games, experiments, and challenges. These activities are designed to be interactive and hands-on, allowing students to explore and discover concepts on their own. They often involve real-life examples and scenarios that students can relate to and apply their knowledge to.

How can inductive playgrounds be incorporated into the classroom?

Inductive playgrounds can be incorporated into the classroom in a variety of ways. Teachers can use them as a supplement to traditional instruction, by introducing hands-on activities and examples to reinforce concepts taught in class. They can also be used as a standalone teaching method, where students explore and discover concepts on their own. Additionally, inductive playgrounds can be used as a form of assessment, where students demonstrate their understanding through practical application and problem-solving.

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