Exercise 3.4.5. Let A be a set. Identify the properties of the relations R and R on (A) described below.
a. For X, Y ∈ A, X R Y if and only if X = Y.
b. For X, Y ∈ A, X R Y if and only if X ≠Y.
Exercise 3.4.6. Consider the relations on the set {1, 2, 3, 4} below.
i. Which are equivalence relations?
ii. For those that are not, determine which properties of an equivalence relation are lacking.
iii. Again, for those that are not, what ordered pairs do you need to add to make them an equivalence relation?
a. {1, 1, 2, 2, 3, 3, 4, 4}
b. {1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 3, 3}
c. {1, 1, 2, 2, 2, 3, 3, 2, 3, 4, 4}