Question

.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.

Answer #1

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 |.

one of your curious students noted the following interesting
relationship; when she added 2 consecutive odd integers, teh sum
was divisible by 2. when she added 3 consecutive odd integers the
sum was divisible by 3. she made the conjecture that the sum of n
consecutive odd numbers is divisible by n. is she correct? how
would you help her prove or disprove her conjecture

Prove the following statements by contradiction
a) If x∈Z is divisible by both even and odd integer, then x is
even.
b) If A and B are disjoint sets, then A∪B = AΔB.
c) Let R be a relation on a set A. If R = R−1, then R is
symmetric.

1.
Let A and B be sets. The set B is of at least the same size as
the set A if and only if (mark all correct answers)
there is a bijection from A to B
there is a one-to-one function from A to B
there is a one-to-one function from B to A
there is an onto function from B to A
A is a proper subset of B
2.
Which of these sets are countable? (mark all...

Prove the following statements:
1- If m and n are relatively prime,
then for any x belongs, Z there are integers a; b such that
x = am + bn
2- For every n belongs N, the number (n^3 + 2) is not divisible
by 4.

Prove the following using induction:
(a) For all natural numbers n>2, 2n>2n+1
(b) For all positive integersn,
1^3+3^3+5^3+···+(2^n−1)^3=n^2(2n^2−1)
(c) For all positive natural numbers n,5/4·8^n+3^(3n−1) is
divisible by 19

1. Prove that an integer a is divisible by 5 if and only if a2
is divisible by 5.
2. Deduce that 98765432 is not a perfect square. Hint: You can use
any theorem/proposition or whatever was proved in class.
3. Prove that for all integers n,a,b and c, if n | (a−b) and n |
(b−c) then n | (a−c).
4. Prove that for any two consecutive integers, n and n + 1 we
have that gcd(n,n + 1)...

Using field axioms and order axioms prove the following
theorems
(i) The sets R (real numbers), P (positive numbers) and [1,
infinity) are all inductive
(ii) N (set of natural numbers) is inductive. In particular, 1
is a natural number
(iii) If n is a natural number, then n >= 1
(iv) (The induction principle). If M is a subset of N (set of
natural numbers) then M = N
The following definitions are given:
A subset S of R...

Prove or disprove each of the following statements:
(a) For all integers a, a | 0.
(b) For all integers a, 0 | a.
(c) For all integers a, b, c, n, and m, if a | b and a | c, then
a | (bn+cm).

Prove the statement in problems 1 and 2 by doing the following
(i) in each problem used only the definitions and terms and the
assumptions listed on pg 146, not by any previous establish
properties of odd and even integers (ii) follow the direction in
this section (4.1) for writing proofs of universal statements
for all integers n if
n is odd then n3 is odd
if a is any odd
integer and b is any even integer, then 5a+4b...

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