MHB Using isomorphism and permutations in proofs

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Using isomorphism and permutations in combinatorial proofs can be challenging, particularly in determining when to apply "without loss of generality" (WLOG). WLOG is a technique that simplifies proofs by allowing the focus on a specific case, as other cases can be derived similarly. The discussion highlights the need for guidelines on utilizing symmetry effectively in arguments. An example is provided regarding the proof that any (7, 7, 4, 4, 2)-designs must be isomorphic. Understanding these concepts is crucial for advancing in combinatorial proofs.
annie122
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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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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<<.
 

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