Problem 6 Let A and B be sets, and let f: A → B be a function. Let ∼ be the relation on A defined by x ∼ y if and only if f (x) = f (y), for all x, y ∈A. Prove that ∼ is an equivalence relation.

7. Let A and B be sets, and let f: A → B and g: B –→ A be functions. Suppose that: • go f(a) = a for every a E A; fo g(b) : = b for every b E B. Prove that f is bijective.

Define an equivalence relation on R² by: ¤1, Y1) ~ (x2, Y2) E if and only if (x1, Y1) and (x2, Y2) have the same distance from the origin. (a) Interpret the equivalence classes geometrically and describe the quotient R?/ ~. (b) Is the function f: (R²/ ~) → R, [(x1,y1)] → xỉ + yỉ well defined?

Let S be the set of all students at FIU. Determine whether the relation R on S (defined below) is reflexive, irreflexive, symmetric, antisymmetric and transitive? Support your answer with a proof or a counterexample per each property. R = {(x, y) = SxSx has a better GPA than y} alternatively, we can write Ry if student x has a better GPA than student y