Question

Let N denote the set of positive integers, and let x be a number which does not belong to N. Give an explicit bijection f : N ∪ x → N.

Answer #1

3. Let N denote the nonnegative integers, and Z denote the
integers. Define the function g : N→Z defined by g(k) = k/2 for
even k and g(k) = −(k + 1)/2 for odd k. Prove that g is a
bijection.
(a) Prove that g is a function.
(b) Prove that g is an injection
. (c) Prove that g is a surjection.

Prove that for fixed positive integers k and n, the number of
partitions of n is equal to the number of partitions of 2n + k into
n + k parts.
show by using bijection

Let E = {0, 2, 4, . . .} be the set of non-negative even
integers
Prove that |Z| = |E| by defining an explicit bijection

Let n be a positive integer, and let Hn denote the graph whose
vertex set is the set of all n-tuples with coordinates in {0, 1},
such that vertices u and v are adjacent if and only if they differ
in one position. For example, if n = 3, then (0, 0, 1) and (0, 1,
1) are adjacent, but (0, 0, 0) and (0, 1, 1) are not. Answer the
following with brief justification (formal proofs not
necessary):
a....

3. Let T be the set of integers that are not divisible
by 3. Prove that T is a countable set by finding a bijection
between the set T and the set of integers Z, which we know is
countable from class. (You need to prove that your function is a
bijection.)

Characterize the set of all positive integers n for which φ(n)
is divisible by 2 but not by 4

Let X Geom(p). For positive integers n, k define
P(X = n + k | X > n) = P(X = n + k) / P(X > n) :
Show that P(X = n + k | X > n) = P(X = k) and then briefly
argue, in words, why this is true for geometric random
variables.

Euler's Totient Function
Let f(n) denote Euler's totient function; thus, for a positive
integer n, f(n) is the number of integers less than n which are
coprime to n. For a prime p its is known that f(p^k) = p^k-p^{k-1}.
For example f(27) = f(3^3) = 3^3 - 3^2 = (3^2) 2=18. In addition,
it is known that f(n) is multiplicative in the sense that
f(ab) = f(a)f(b)
whenever a and b are coprime. Lastly, one has the celebrated
generalization...

Let
m and n be positive integers. Exhibit an arrangement of the
integers between 1 and mn which has no increasing subsequence of
length m + 1, and no decreasing subsequence of length n + 1.

Let (X, d) be a metric space, and let U denote the set of all
uniformly continuous functions from X into R. (a) If f,g ∈ U and we
define (f + g) : X → R by (f + g)(x) = f(x) + g(x) for all x in X,
show that f+g∈U. In words,U is a vector space over R. (b)If f,g∈U
and we define (fg) : X → R by (fg)(x) = f(x)g(x) for all x in X,...

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