Let f: X→Y be a surjective function. Define a relation on X by setting ay iff f(x) = f(y). Let E be the set of equivalence classes. Show that there is a bijection from E to Y.

Let f x 1

Let A and B be two nonempty sets, f: A B be a function and R be an equivalence relation on the set B. For x1, x2 e A, we put xRx2 if and only if f(x1)Rf(x2). Show that R' is an equivalence relation on the set A.

Let A = {1, 3} and B = {2, 4, 6}. Define a relation R from A to B as follows:for all (x, y) ∈ A × B (x, y) ∈ R if and only if y > x + 1 (a) List all ordered pairs in R.(b) Draw an arrow diagram of R.(c) Is R a function? Support your answer.

Decide whether the following relations is an equivalence relation or not. {(x, y) | xy > 0, x and y are integers}

Let A = {(x,y) |x-yeQ for x, y = R} be a relation on R. Which statement best describes problem? A Reflexive (Yes), symmetric (Yes), transistive (Yes) Reflexive (No), symmetric (No), transistive (Yes) Reflexive (Yes), symmetric (No), transistive (No) Reflexive (No), symmetric (No), transistive (No) Reflexive (No), symmetric (Yes), transistive (No) B …….

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 f: A → B be a surjective function. We define the relation ~ on A by x ~ y ⇒ f (x) = f(y). (a) Prove that is an equivalence relation. (b) Find a bijection from A/~ to B.

Let f be a function from X toY. Define a relation R on X by xRy    if f(x) = f(y) Show that R is an equivalence relation. Is there a natural one-to-one function g related to f that maps the set of equivalence classes of R to Y ?

Discrete Math