Question

Let E = {0, 2, 4, . . .} be the set of non-negative even integers

Prove that |Z| = |E| by defining an explicit bijection

Answer #1

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.)

Define the set E to be the set of even integers; that is,
E={x∈Z:x=2k, where k∈Z}. Define the set F to be the set of integers
that can be expressed as the sum of two odd numbers; that is,
F={y∈Z:y=a+b, where a=2k1+1 and
b=2k2+1}.Please prove E=F.

4. Let Z be the set of all integers (positive, negative and
zero.) Write a
sequence containing every element of Z.

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.

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: Let a and b be integers. Prove that integers a and b are
both even or odd if and only if 2/(a-b)

Let a, b be an element of the set of integers. Proof by
contradiction: If 4 divides (a^2 - 3b^2), then a or b is even

Find the cardinality of the following sets:
(d) S={n ∈ N(natural) | n is even} ← prove! write a
bijection.
(e) S = Z(integers) ← prove! write a bijection.

Prove that the cardinality of of 2Z (the set of even integers)
is ℵ0.

Suppose the sum of some set X of 5 non-negative integers is 51.
Show that there must be a subset of four of them with sum at least
41.

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