For a b c d restricted to the universe of positive integers

For a, b, c, d restricted to the universe of positive integers, explain why a b c d a/b = c/d is true, but a d c b a/b = c/d is false.

Solution

1] a b c d a/b = c/d

what this mean is that if you take any value of a ( and you have to repeat this \"any value\" differently each time. for eg take a= 3 1st time, 9 2nd time and so on but for now stick with one value so, 3), then there exists b , this b will make the ratio of a/b fix. (we can easily fix this ratio, for example for any value of a , there exists a value of b so that I can make this ratio equal to 1). for all c there exists d such that this ratio of 1 can be attained as well. so 1 is true.

2] a d c b a/b = c/d

lets take a= x (which represents a general value like it normally means) there exist d (say that d=x_0). so, only way to make this equality work is equalling the product ad = bc . Now, here we have chose ad. Lets move forward. For all c (this is where the statement gets wrong, I can take C = 100, and ab =50 so , there does not exist any d such that relation hold. So, it is false.