Question

Assume that X and Y are finite sets. Prove the following
statement:

If there is a bijection f:X→Y then|X|=|Y|.

Hint: Show that if f : X → Y is a surjection then |X| ≥ |Y| and if
f : X → Y is an injection then

|X| ≤ |Y |.

Answer #1

X，Y be sets and f:X-＞Y is a function there's a function g:Y-＞X
such that g(f(x))＝x for all x∈X
Prove or disprove: f is a bijection

.Unless otherwise noted, all sets in this module are finite.
Prove the following statements!
1. There is a bijection from the positive odd numbers to the
integers divisible by 3.
2. There is an injection f : Q→N.
3. If f : N→R is a function, then it is not surjective.

Prove that if X and Y are disjoint countably infinite sets then
X ∪ Y is countably infinity (can you please show the bijection from
N->XUY clearly)

3. Prove or disprove the following statement: If A and B are
finite sets, then |A ∪ B| = |A| + |B|.

Let A be a finite set and let f be a surjection from A to
itself. Show that f is an injection.
Use Theorem 1, 2 and corollary 1.
Theorem 1 : Let B be a finite set and let f be a function on B.
Then f has a right inverse. In other words, there is a function g:
A->B, where A=f[B], such that for each x in A, we have f(g(x)) =
x.
Theorem 2: A right inverse...

Suppose that f is a bijection and f ∘ g is defined. Prove:
(i). g is an injection iff f ∘ g is;
(ii). g is a surjection iff f ∘ g is.

Please prove the following theorem:
Suppose (X,p) and (Y,b) are metric spaces, X is compact, and
f:X→Y is continuous.
Then f is uniformly continuous.

Is f(x) = x^3 - x an injection, surjection, or bijection? How do
I know?

Let X and Y be sets. Prove X −(X −Y ) ⊆ X ∩Y . (Hint: Remember
that s ∈ S − T means s ∈ S and s ∈/ T . Thus, s ∈/ S − T means s ∈/
S or s ∈ T .)

Multiplicative Principle (in terms of sets): If X and Y are
ﬁnite sets, then |X ×Y| = |X||Y|.
D) You are going to give a careful proof of the multiplicative
principle, as broken up into two steps:
(i) Find a bijection
φ : <mn> → <m> × <n>
for any pair of natural numbers m and n. Note that you must
describe explicitly a function and show it is a bijection.
(ii) Give a careful proof of the multiplicative principle...

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