Question

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 injective and g is surjective.

Answer #1

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:A→B and g:B→C be maps. Prove that if g◦f is a bijection,
then f is injective and g is surjective.*You may not use, without
proof, the result that if g◦f is surjective then g is surjective,
and if g◦f is injective then f is injective. In fact, doing so
would result in circular logic.

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

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.

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 f : A → B and g : B → C. For each statement below either
prove it or construct f, g, A, B, C which show that the statement
is false.
(a) If g ◦ f is surjective, then g is surjective.
(b) If g ◦ f is surjective, then f is surjective.
(c) If g ◦ f is injective, then f and g are injective

let G be a group of order 18. x, y, and z are elements
of G. if | < x, y >| = 9 and o(z) = order of z = 9, prove
that G = < x, y, z >

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.

Prove: Let x,y be in R such that x < y.
There exists a z in R such that x < z <
y.
Given:
Axiom 8.1. For all x,y,z in
R:
(i) x + y = y + x
(ii) (x + y) + z = x + (y + z)
(iii) x*(y + z) = x*y + x*z
(iv) x*y = y*x
(v) (x*y)*z = x*(y*z)
Axiom 8.2. There exists a real number 0 such that
for all...

Let G be a graph with x, y, z є V(G). Prove that if G contains
an x, y-path and a y, z-path, then it contains an x, z-path.

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