On the set X = {1,2,3,4,5,6,7,8,9}, there is binary relation Q = {(1,9),(2,5),(3,7),(4,1),(5,8),(6,2),(7,3),(8,6),(9,4)}. Make a transitive closure T of the relation Q. Decide and prove whether the relation T is an equivalence on X.
If yes, write out the partition belonging to this equivalence. If not, find the partition belonging to the least equivalence which contains the relation T. Also state how many classes does the partition have.
Could you please help me understand the way this is solved?
This is the solution I have in my study materials:
Q = {(1,9),(9,4),(4,1),(2,5),(5,8),(8,6),(6,2),(3,7),(7,3)}
Q2 = {(1,4),(9,1),(4,9),(2,8),(5,6),(8,2),(6,5),(3,3),(7,7)}
Q3 = {(1,1),(4,4),(9,4),(6,8),(2,6),(5,2),(8,5),(3,7),(7,3)}
Q4 = {(1,9),(9,4),(4,1),(2,2),(5,5),(6,6),(8,8),(3,3),(7,7)}
T = {(1,1),(4,4),(4,9),(1,4),(1,9),(4,1),(4,9),(9,1),(9,4),(2,2),(5,5),(6,6),(8,8),(2,8),(2,5),(2,6),(5,2),(5,6),(5,8),(6,2),(6,5),(6,8),(8,2),(8,5),(8,6),(3,3),(7,7),(3,7),(7,3)} = T-1
Which means T is symmetric and reflexive, because X = {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(7,7),(8,8),(9,9)}. As a transitive closure T is transitive -> therefore it is an equivalence.