Using isomorphism and permutations in proofs

In summary, the conversation discusses the use of isomorphism and permutation in combinatorics proofs, the concept of "without loss of generality", and guidelines for using symmetry in arguments. There is also mention of a specific problem involving symmetry in a proof.
  • #1
annie122
51
0
I have trouble using isomorphism and permutation in proofs for combinatorics.
I don't know when I can assume "without loss of generality".
What are some guidelines to using symmetry in arguments.

One problem I'm working on that uses symmetry is to "prove that any (7, 7, 4, 4, 2)-designs must be isomorphic."
 
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  • #2
Yuuki said:
I don't know when I can assume "without loss of generality".

Hi Yuuki, :)

Well, I don't have the background to answer all your questions but would like to clarify about statement "without loss of generality". First of all it's not an assumption but rather a way of narrowing down a proof to a special case. All the other possibilities of the proof just follows the same procedure with symbols and objects interchanged. You would find a nice example about the use of this statement >>here<<.
 

Related to Using isomorphism and permutations in proofs

1. What is isomorphism and how is it used in proofs?

Isomorphism is a mathematical concept that refers to the idea of two structures being similar in some way. In proofs, isomorphism is used to show that two objects or structures have the same properties and can be transformed into each other without changing their fundamental nature.

2. How can permutations be helpful in proving isomorphism?

Permutations are used in proofs to show that the elements of two structures can be rearranged in the same way, indicating that they are isomorphic. By comparing the permutations of two structures, we can determine if they have the same underlying structure and are therefore isomorphic.

3. Can isomorphism and permutations be used in any type of mathematical proof?

Yes, isomorphism and permutations can be used in a variety of mathematical proofs, particularly in abstract algebra, group theory, and graph theory. These concepts help to prove that two structures are equivalent and can be manipulated in the same way.

4. How do you determine if two structures are isomorphic using permutations?

To determine if two structures are isomorphic using permutations, you need to compare the elements of each structure and see if they can be rearranged in the same way. If the elements can be arranged in the same way, then the structures are isomorphic.

5. Can isomorphism and permutations be applied to real-world problems?

Yes, isomorphism and permutations can be applied to real-world problems, particularly in the fields of computer science and chemistry. For example, isomorphism can be used in computer science to compare the structures of different data sets, while permutations can be used in chemistry to study the properties of different molecules.

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