1*) Let X = {1,2, 3}, Y = {2,3, 4}, and Z = {1,2}.(a) Define a function f: Z → X that is one-to-one but not onto.(b) Define a function g: X → Z that is onto but not one-to-one.(c) State whether there exists a function h: X →Y that is one-to-one but not onto.%3D2) (a) Let f(x) = 31 for all real numbers r 0. Prove or disprove that f is one-to-one.'(b*) Let f(x) = 1+ Vx – 2 for all real numbers x > 2. Prove or disprove that f is one-to-one.(c*) Let f(x) = i for all real numbers x. Prove or disprove that f is one-to-one.3*) Suppose f : Q → Z is defined bym= m – n for all integers m and n with n0.Is f a well defined function?4*) Let Pf(Z) denote the set of all finite subsets of Z.Let f: {n E Z |n > 2} → Pf(Z) be the function defined byf(n)= the set of prime divisors of n,for all integers n > 2.(a) Find f(2), f(24), ƒ(27), and f(30).(b) Is ƒ(n) a nonempty set for all integers n > 2? Why?5*) Suppose f : R → R is a function and c is a real number. A new function denoted cf, wherecf: R → R, can be defined by (cf)(x)f: R → R and all nonzero real numbers c, if f is onto, then cf is onto.= c • f(x) for all x e R. Show that for all functions6) Let f: R × R → R × R be defined by f(x, y) = (x+1, 2y) for all (x, y) E R × R. Prove thatf is one-to-one and onto.7) Suppose f: X → Y is a one-to-one and onto function. Prove that f-1: Y → X is alsoone-to-one and onto.8*) Suppose f: X →Y is an onto function. Prove that for all subsets BCY, BC f[f¯'[B]].?9*) Suppose f: X → Y is a one-to-one function. Prove that for all subsets A and B of X,f[A]n f[B] C f[An B].³10*) Suppose f: X → Y is any function. Prove that for all subsets C and D of Y , f-'[C– D] =f-"|C) – f-'[D].11*) For a fixed set S and any subset A of S, the characteristic function of A, denoted XA, isthe function from S to {0,1} defined by(1 if x € AXA(x) =for all x E S.if x ¢ A,Prove that for all subsets A and B of S and for all æ E S,XAUB (T) = XA(x) + XB(x) – Xa(x) · XB(x).%3DIThe codomain, which is inessential, is understood to be R when it is not explicitly stated.2The backward inclusion is always true, regardless of whether f is surjective or not. Can you prove this?3The backward inclusion is always true, regardless of whether ƒ is injective or not. Can you prove this?

Note: According to our guidelines We’ll answer the first question since the exact one wasn’t…

1*) Let X = {1,2, 3}, Y = {2,3, 4}, and Z = {1,2}. (a) Define a function f: Z→ X that is one-to-one but not onto. (b) Define a function g: X Z that is onto but not one-to-one. (c) State whether there exists a function h: X →Y that is one-to-one but not onto. 2) (a) Let f(x) = -1 for all real numbers r 0. Prove or disprove that f is one-to-one.' (b*) Let f(x) = 1+vx – 2 for all real numbers r > 2. Prove or disprove that f is one-to-one. (c*) Let f(x) = i for all real numbers r. Prove or disprove that f is one-to-one. %3D 3*) Suppose f: Q → Z is defined by = m – n for all integers m and n with n # 0. Is f a well defined function? 4*) Let Pf(Z) denote the set of all finite subsets of Z. Let f: {n e Z|n > 2} → P;(Z) be the function defined by f(n) = the set of prime divisors of n, for all integersn2 2. (a) Find f(2), f(24), ƒ(27), and f (30). (b) Is f(n) a nonempty set for all integers n > 2? Why? 5*) Suppose f: R → R is a function and c is a real number. A new function denoted cf, where ef : R → R, can be defined by (cf)(x) = c · f(æ) for all x e R. Show that for all functions f: R → R and all nonzero real numbers c, if f is onto, then cf is onto. 6) Let f: R x R → R × R be defined by f(x, y) = (x +1, 2y) for all (x, y) E R × R. Prove that f is one-to-one and onto. 7) Suppose f: X → Y is a one-to-one and onto function. Prove that f-1: Y → X is also one-to-one and onto. 8*) Suppose f: X →Y is an onto function. Prove that for all subsets BCY, BC f[f¯'[B]].² 9*) Suppose f: X → Y is a one-to-one function. Prove that for all subsets A and B of X, f[A]n f[B] C f[An B].³ 10*) Suppose f: X → Y is any function. Prove that for all subsets C and D of Y, ƒ-'[C– D] = f-|C) – f-"[D]. 11*) For a fixed set S and any subset A of S, the characteristic function of A, denoted XA, is the function from S to {0, 1} defined by J1 if a E A l0 if x ¢ A, XA(x) = for all r E S. Prove that for all subsets A and B of S and for all a E S, XAUB (T) = XA(x) +XB(x) – Xa(x) · XB(x). IThe codomain, which is inessential, is understood to be R when it is not explicitly stated. 2The backward inclusion is always true, regardless of whether f is surjective or not. Can you prove this? 3The backward inclusion is always true, regardless of whether f is injective or not. Can you prove this?