Question

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.

Answer #1

Let a function by

**a)**

** **

**b) f is injective (one-to-one).**

if

Which implies
which implies **f is injective**

**c) f is not surjective (onto)**

for any

If there there n such that

where 3 is not square

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

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...

Let f : R → R + be defined by the formula f(x) = 10^2−x . Show
that f is injective and surjective, and find the formula for f −1
(x).
Suppose f : A → B and g : B → A. Prove that if f is injective
and f ◦ g = iB, then g = f −1 .

Let X = [0, 1) and Y = (0, 2).
a. Define a 1-1 function from X to Y that is NOT onto Y . Prove
that it is not onto Y .
b. Define a 1-1 function from Y to X that is NOT onto X. Prove
that it is not onto X.
c. How can we use this to prove that [0, 1) ∼ (0, 2)?

If f:X→Y is a function and A⊆X, then define
f(A) ={y∈Y:f(a) =y for some a∈A}.
(a) If f:R→R is defined by f(x) =x^2, then find f({1,3,5}).
(b) If g:R→R is defined by g(x) = 2x+ 1, then find g(N).
(c) Suppose f:X→Y is a function. Prove that for all B,
C⊆X,f(B∩C)⊆f(B)∩f(C). Then DISPROVE that for all B, C⊆X,f(B∩C)
=f(B)∩f(C).

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...

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

Consider the cubic function f: x^3 -10x^2 - 123x + 432
a) Sketch the graph of f.
b) Identify the domain and co-domain in which f is NOT Injective
and NOT surjective. Briefly explain
c) Identify the domain and co-domain in which f is Injective but
NOT Surjective. Briefly explain
d) Identify the domain and co-domain in which f is Surjective
but NOT Injective. Briefly explain
e). Identify the domain and co-domain in which f is Bijective.
Explain briefly

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.

1. Let a, b ∈ Z. Define f : Z → Z by f(n) = an + b. Prove that f
is one to one if and only if a does not equal 0.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 22 seconds ago

asked 19 minutes ago

asked 35 minutes ago

asked 55 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago