HOME WORK
Question One:
The function f: ℝ → ℝ is defined by the rule f(x) = 2×2 – 3x + 1 .
Show that f is not a one-to-one function and that it is not an onto function.
- Which interval of ℝ is the range of the function f ?
- Express the image set f([1,2]) as an interval of ℝ .
- Express the preimage set f_1([0, ∞)) as a union of intervals of ℝ .
Question Two:
Let h: A → B be a one-to-one function from a set A to a set B. The function f can be used to construct a function H:(A) → (B) as follows: For each X ∈ (A)
H(X) = h(X) (i.e. the image set of , by ℎ ). Show that is one-to-one ⇔ ℎ is one-to-one.
Question Three:
Let A , B and C be subsets of a universal set. Either prove or provide a counter example for the following statements:
(A ∖ (B ∩ C)) ∖ (B ∖ C) = A ∖ B .
(A ∖ B) ∖ (B ∖ C) = A ∖ C.
(A ∪ B) ∖ C = A ∪ (B ∖ C).
Question Four:
Show that the function f: ℕ × ℕ → ℕ , defined by the rule
f(m.n) = 2m3n is a one-to-one function (injective) function.
Show that the function g: ℝ ∖ {3} → ℝ ∖ {2} , defined by the rule
g(x) = 2x
x~3
is a one-to-one function (injective) function, which is onto. Find the inverse function g_1 of the function g .
Question Five:
The function f: (0,1) → (0,1) is defined by the rule f(x) = 2x
1+y2 . Show that f is a one-to-one and onto function. Find the rule f_1(x) of the inverse function f~1