Question

Let f: Z -> Z be a function given by f(x) = ⌈x/2⌉ + 5. Prove that f is surjective (onto).

Answer #1

2. Define a function f : Z → Z × Z by f(x) = (x 2 , −x).
(a) Find f(1), f(−7), and f(0).
(b) Is f injective (one-to-one)? If so, prove it; if not,
disprove with a counterexample.
(c) Is f surjective (onto)? If so, prove it; if not, disprove
with a counterexample.

8.4: Let f : X → Y and g : Y→ Z be maps. Prove that if
composition g o f is surjective then g is surjective.
8.5: Let f : X → Y and g : Y→ Z be bijections. Prove that if
composition g o f is bijective then f is bijective.
8.6: Let f : X → Y and g : Y→ Z be maps. Prove that if
composition g o f is bijective then f is...

Let X, Y and Z be sets. Let f : X → Y and g : Y → Z functions.
(a) (3 Pts.) Show that if g ◦ f is an injective function, then f is
an injective function. (b) (2 Pts.) Find examples of sets X, Y and
Z and functions f : X → Y and g : Y → Z such that g ◦ f is
injective but g is not injective. (c) (3 Pts.) Show that...

5. Prove that the mapping given by f(x) =x^3+1 is a function
over the integers.
6. Prove that f(x) =x^3+is 1-1 over the integers
7. Prove that f(x) =x^3+1 is not onto over the
integers
8 Prove that 1·2+2·3+3·4+···+n(n+1)
=(n(n+1)(n+2))/3.

. Let f : Z → N be function.
a. Prove or disprove: f is not strictly increasing. b. Prove
or disprove: f is not strictly decreasing.

Let f: Z→Z be the functon defined by f(x)=x+1. Prove that f is a
permutation of the set of integers. Let g be the permutation (1 2 4
8 16 32). Compute fgf−1.

Consider the function f : Z → Z defined by f(x) = x 2 . Is this
function one-to-one, onto, or neither? Give justification for your
claims that rely on definitions.
With explanation please

Let A be a non-empty set and f: A ? A be a function.
(a) Prove that, if f is injective but not surjective (which
means that the set A is infinite), then f has at least two
different left inverses.

Problem 2. Let F : R
→ R be any function (not necessarily measurable!).
Prove that the set of points x ∈ R such
that
F(y) ≤ F(x) ≤
F(z)
for all y ≤ x and z ≥ x is
Borel set.

Let f : R − {−1} →R be defined by f(x)=2x/(x+1).
(a)Prove that f is injective.
(b)Show that f is not surjective.

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