QuestionQuestion

1. Find all functions from X = {a, b, c} to Y = {u, v}.
2. Let F and G be functions from the set of all real numbers to itself. Define new functions F – G: R -> R and G - F: R -> R as follows:
(F – G)(x) = F(x) – G(x) for all x ϵ R,
(G – F)(x) = G(x) – F(x) for all x ϵ R.
Does F – G = G – F? Explain.
3. Let A = {2, 3, 5} and B = {x, y}. Let p1 and p2 be the projections of AxB onto the first and second coordinates. That is, for each pair (a, b) ϵ AxB, p1(a, b) = a and p2(a, b) = b.
Find p2(2, y) and p2(5, x). What is the range of p2?
4. Let X = {1, 5, 9} and Y = { 3, 4, 7}. Define g : X->Y by specifying that g(1) = 7, g(5) = 3, g(9) = 4. Is g one-to-one? Is g onto? Explain your answers.
5. Let X = {1, 2, 3}, Y = {1, 2, 3, 4}, and Z = {1, 2}.
a) Define a function g : X -> Z that is onto but not one-to-one.
b) Define a function k : X -> X that is one-to-one and onto but is not the identity function on X.
6. Let X = {1, 2, 3, 4}, Y = {2, 3, 4, 5, 6}, Z = {1, 2, 3}.
a) Define a function f: X -> Y that is one-to-one but not onto.
b) Define a function g: X -> Z that is onto but not one-to-one.
c) Define a function h: X -> Y that is neither onto nor one-to-one.
d) Define a function k: X -> X that is onto and one-to-one but is not the identity function on X.
7. List all the functions from the three element set {1, 2, 3} to the set {a, b}. Which functions, if any, are one-to-one? Which functions, if any, are onto?
8. Define f: R -> R by the rule f(x) = 2x² -3x+1
a) Is f one-to-one? Prove or give a counterexample.
b) Is f onto? Prove or give a counterexample.
9. Define g : Z -> Z by the rule g(n) = 3n – 2, for all integers n.
a) Is g one-to-one? Prove or give a counterexample.
b) Is g onto? Prove or give a counterexample.
10. Let X = {a, b, c, d, e} and Y = {s, t, u, v, w}. A one-to-one correspondence F: X-> Y is defined by: F(a) = t, F(b) = w, F(c) = s, F(d) = u, F(e) = v. Define F-¹ (Please, specify each function value, i.e. F-¹(s) = ...what-ever, or draw the diagram).

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4) Yes, the defined function is one-to-one because for xi and xj from X we have g(xi)=g(xj)=> xi=xj. Different values from X are mapped to different values from Y.
Also the function g is onto because (∀) y ∈ Y (∃) x ∈ X such that g(x)=y (with no exception)....

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