Suppose f is a function from a set A to its power set P(A). Colour an element x
ID: 1889315 • Letter: S
Question
Suppose f is a function from a set A to its power set P(A). Colour an element x of A blue if x is in f(x). Colour all the other elements red. Let R be the subset of A made up of red elements only.(a) For the set A = {1,2,3} list the elements of P(A).
(b) Suppose f(1) = {1,2}, f(2) = {1}, and f(3) = {1,2,3}. In this example, what are the elements of R?
(c) (in general, not in the example of parts a and b): Prove that there is no element x is in A with f(x) = R.
I have an answer, but I would like to double check my work and answer. So please wok out the problem in steps. Thank you.
Explanation / Answer
a) The power set of A is simply the set of all subsets of A, so P(A) = {empty set, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} b) Since 1 is in f(1) it's blue. Since 2 is NOT in f(2) it's red. Since 3 is in f(3) it's blue. Therefore the blue elements are 1 and 2, whereas the red element is 2. So R = {2}. c) Assume there is an x in A with f(x) = R. If x is in A with f(x) = R, then that would make x blue contradicting the fact that it's also in R. Therefore there is no element x in A with f(x) = R.