an isomorphism is like a bijection, it is basically 2 things which are basically the same question but just worded differently. i used the word very casually here -_- perhaps i shud have said bijection but anyways it was pretty informal
the one i've always remembered when i learnt combinatorial arguments is this example:
prove the summation identity
for
one could do a algebraic 'proof' but one can also avoid the use of any rigorous/formal maths and simply try an "combinatorial argument" like this:
say n = 17 and r = 10 then we have:
now my combinatorial argument will be like this:
Let us consider all 11 member committees formed from a group of 18 people, fix one of the 18 people, say some one called E. The 11 member committees can be broken down into two mutually exclusive types: those with Eand those without. How many include E? Having already chosen E, we are free to chose 10 more people from the remaining pool of 17, so we got:
committees that include E. To count the committees without E, we must choose 11 people, but again out of 17, since we need to remove E from the original pool of 18. so this means
committees exclude E. The total number of 11 member committees is the sum of the number of committees with Erika plus the number without Erika, which is
, but this is just the total of the 2 mutually exclusive cases which proves the inequality for this specific case.
Then just replace 17 with n and 10 with r and do the same combinatorial argument thus "proving" the equality.
can u see how a combinatorial argument is like a "story" which basically requires a bit of wishful thinking to create ur OWN problem which yields the result, rather than using known identities to prove the result.
for example, create a combinatorial argument for 7 x 8 = 56
what does it mean? well it could mean that there are 7 choices of pasta and 8 choices of pizza so there are 56 ways of choosing a pasta dinner with pizza.
other types of isomorphisms or "bijections" could be like generating functions and combinatorics
you could have a very complex combinatorics questions but simply represent it with an expression like (1+x^2+x^4+...)(1+x^3+x^5+...) or something whose indices could represent different combinations.
another type of basic isomorphic relationship could be in basic linear algebra, where you can view a matrix using different methods such as using the column picture of a matrix or a row picture, both yield different equations but when solved, they represent the exact same thing.
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