The discussion focuses on determining the subset Yf for the function f defined as f:X->P(X), where f(x) = X/x. Participants clarify that for a set X={1,2,3}, Yf includes all elements of X except the element x being evaluated. Specifically, Yf = {x in X such that x not in f(x)}, leading to the conclusion that Yf equals X for any non-empty set X. The conversation emphasizes understanding the notation and the implications of the function definition.
PREREQUISITESMathematics students, educators, and anyone interested in set theory and function analysis will benefit from this discussion.
hocuspocus102 said:f(1) in that case would be {2,3} right? but how does that translate to Yf?
hocuspocus102 said:no, so is Yf just X/x ?
hocuspocus102 said:no 1 isn't in it because it's x in X such that x not in f(x) and 1 is not in f(1)
hocuspocus102 said:I get that for f(1), 2 is in Y_f and 3 is in Y_f but 1 is not.
for f(2), 1 is in Y_f and 3 is in Y_f but 2 is not.
so for f(n) on a set of size t Y_f will have all numbers in the set 1 through t not including n in it. is that how you'd say that or is there a more explicit formula?
hocuspocus102 said:I don't know I thought f just took n to a set with all elements of X not including n because Y_f is x not in f(x)
hocuspocus102 said:no it's not
hocuspocus102 said:ohhhh. ok so Y_f is whatever value you plug in for x?
hocuspocus102 said:{1,2,3} are in Y_f because f(1) implies 1 is in it, f(2) implies 2 is in it and f(3) implies 3 is in it right?
hocuspocus102 said:it should
hocuspocus102 said:so Y_f is just X?
hocuspocus102 said:that's what I think