University math proofs. Work on paper and show every step

1. For this question, recall the interval notation for subsets of R: If a, b are real numbers with a 0 such that for all r E A, (x-r, Itr) C A. . Sofia likes A if and only if for every r E A there exists an r > 0 such that (x-r, rtr) CA. . Themba likes A if and only if for every r ( A there exists anr > 0 such that [x-r, str] C A. . Yash likes A if and only if there exists an r > 0 and there exists an r E A such that [x-r,atr] CA. Be sure to prove the correctness of your answers to the following questions. (a) Identify which TAs, if any, like the empty set. (b) Give an example of a set A which every TA dislikes. (c) Show that Sofia and Themba like exactly the same sets. (d) Determine whether the following statement is true: "For every A C R, if Yash likes A then Sofia likes A.&quot