Let A = 1, 2, 3, 4, 5, 6 . A function f: A A is defined by f(1)=3 , f(2)=2 , f(3)=4 , f(4)=1 , f(5)=6 , and f(6)=5 . A relation R on A is defined as R = (x, f(x)) : x A . The minimum number of elements that must be added to R so that it becomes an equivalence relation is:

Step-by-step solution

First, we list the ordered pairs in the relation R based on the given function f : R = (1,3), (2,2), (3,4), (4,1), (5,6), (6,5) To make R an equivalence relation, it must be reflexive, symmetric, and transitive. This is equivalent to finding the equivalence classes that contain the connected components of R . Looking at the pairs, we can trace the cycles formed by the elements: - 1 3 4 1 (Elements 1, 3, 4 must be in the same equivalence class) - 2 2 (Element 2 forms its own class) - 5 6 5 (Elements 5, 6 must be in