Assignment:
Q1. Find all functions from X = {a, b, c} to Y = {u, v}.
Q2. 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 ∈ R,
(G – F)(x) = G(x) – F(x) for all x ∈ R.
Does F – G = G – F? Explain.
Q3. Let A = {2, 3, 5} and B = {x, y}. Let p1 and p2 be the projections of A x B onto the first and second coordinates. That is, for each pair (a, b) ∈ A x B, p1(a, b) = a and p2(a, b) = b.
Find p2(2, y) and p2(5, x). What is the range of p2?
Q4. 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.
Q5. 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.
Q6. 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.
Q7. 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?
Q8. 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.
Q9. 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.
Q10. 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).
Q11. Give a real-world example of a function which is both one to one and onto.
Provide complete and step by step solution for the question and show calculations and use formulas.