Question

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.

Answer #1

Let A be a non-empty set. Prove that if ∼ defines an equivalence
relation on the set A, then the set of equivalence classes of ∼
form a partition of A.

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 X be a non-empty finite set with |X| = n. Prove that the
number of surjections from X to Y = {1, 2} is (2)^n− 2.

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 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 : 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 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 A be a finite set and f a function from A to A.
Prove That f is one-to-one if and only if f is onto.

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

let F : R to R be a continuous function
a) prove that the set {x in R:, f(x)>4} is open
b) prove the set {f(x), 1<x<=5} is connected
c) give an example of a function F that {x in r, f(x)>4} is
disconnected

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