2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f (y). a. Prove that R is an equivalence relation on A. b. Let Ex = fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x 2 Ag to be the collection of all equivalence classes. Prove that the function g : A ! E deÖned by g (x) = Ex is surjective. c. Prove that the function h : E ! B deÖned by h (Ex) = f (x) is injective. d. Show that f = h g.
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f (y). a. Prove that R is an equivalence relation on A. b. Let Ex = fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x 2 Ag...
How do I prove this function is not surjective? 3.) Let f: R-R, f(x)-x2+ x+1 and Show that f is not injective and not surjective Justify that g is bijective and find gt. PIR, Show all the wortky) Not Surtechive: fx) RB Surjective: ye(o,oo) hng (g) 8 gon)-es is bijecelive g(x)-ex+s
need help with proving discrete math HW, please try write clearly and i will give a thumb up thanks!! Let A and be B be sets and let f:A B be a function. Define C Ax A by r~y if and only if f(x)f(y). Prove thatis an equivalence relation on A. Let X be the set of~-equivalence classes of A. L.e. Define g : X->B by g(x) Prove that g is a function. Prove that g is injective. Since g...
Let X be a set with an equivalence relation ∼. Let f : X/ ∼→ Y be a function with domain as the quotient set X/ ∼ and codomain as some set Y . We define a function ˜f, called the lift of f, as follows: ˜f : X → Y, x 7→ f([x]). We define a function Φ : F(X/ ∼, Y ) → F(X, Y ), f 7→ ˜f. (1) Is Φ injective? Give a proof or a...
Let R be the relation on N defined by xRy iff 2 divides x+y. R is an equivalence relation. You do not have to prove that R is an equivalence relation. True or False: 3 ∈ 4/R.
5. Let A = P(R). Define f : R → A by the formula f(x) = {y E RIy2 < x). (a) Find f(2). (b) Is f injective, surjective, both (bijective), or neither? Z given by f(u)n+l, ifn is even n - 3, if n is odd 6. Consider the function f : Z → Z given by f(n) = (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n E Z. Find [n]. equivalence relation in [N, the equivalence class of 3. We defined a relation on sets A B. Prove that this relation is an (In this view, countable sets the natural numbers under this equivalence relation). exactly those that are are 2. Consider the relation E on Z defined by...
Let X, Y be two nonempty sets and let f : X → Y. For a, b X we write a ~ b iff f(a) = f(b). Prove that~is an equivalence relation on X Write lely for the equivalence class of x e X with respect to “~" Express [ely in terms of the function f: Irl, = {re x : f(z') a: b: ?? J. (I d o not want to see ..|x ' = {x"e X : r,...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
Let h : X −→ Y be defined by h(x) := f(x) if x ∈ F g −1 (x) if x ∈ X − F Now we must prove that h is injective and bijective. Starting with injectivity, let x1, x2 ∈ X such that h(x1) = h(x2). Assume x1 ∈ F and x2 ∈ X −F. Then h(x1) = f(x1) ∈ f(F) and h(x2) = g −1 (x2) ∈ g −1 (X − F) = Y...
13 pts) Let R be the relation on R deÖned by xRy means "sin2 (x) + cos2 (y) = 1". Recall the Pythagorean identity: 8u 2 R we have sin2 (u) + cos2 (u) = 1. (a) (9 pts) PROVE that R is an equivalence relation on R. (b) (4 pts) Describe all elements of the (inÖnite) equivalence class [0]. Recall: sin(0) = 0 and cos(0) = 1. 2. (13 pts) Let R be the relation on R defined by...