2. (Symmetric Relations on Finite Sets: ā...the Same Number of Friendsā). For each of the following relations R on A = An:
(a) find the matrix MR of R;
(b) use (a) to determine whether R is symmetric;
(c) if yes, for every a ā A, determine the number
fa = |{b ā A | a R b and b ā a}|
(the number of āfriendsā);
(d) find a pair a, b ā A of distinct elements of A such that fa = fb (which means that a and b have the same number of āfriendsā).
R1 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 4), (3, 1), (3, 3), (4, 1), (4, 2)} on A4;
R2 = {(1, 1), (1, 2), (1, 5), (2, 1), (2, 2), (2, 3), (2, 5), (3, 2), (4, 4), (5, 1), (5, 2)} on A5;
R3 = {(1, 2), (1, 3), (1, 4), (1, 6), (2, 1), (2, 2), (2, 4), (3, 1), (3, 4), (3, 6), (4, 1), (4, 2), (4, 3), (5, 5), (6, 1), (6, 3), (6, 6)} on A6;
R4 = {(1, 3), (1, 7), (2, 2), (2, 4), (2, 5), (3, 1), (3, 5), (4, 2), (4, 4), (4, 7), (5, 2), (5, 3), (6, 6), (7, 1), (7, 4)} on A7.
Present your answers in a table similar to the following table: