Let a and b be rational numbers. As always, prove your answers.
(a) For which choices...
Let a and b be rational numbers. As always, prove your answers.
(a) For which choices of a, b is there a rational number x such
that ax = b? (b) For which choices of a, b is there exactly one
rational number x such that ax = b?
Question 1. State the prove The Density Theorem for Rational Numbers.
Question 2. Prove that irrational numbers are dense in the set of real numbers.
Question 3. Prove that rational numbers are countable
Question 4. Prove that real numbers are uncountable
Question 5. Prove that square root of 2 is irrational
Question 1. State the prove The Density Theorem for Rational Numbers.
Question 2. Prove that irrational numbers are dense in the set of real numbers.
Question 3. Prove that rational numbers are countable
Question 4. Prove that real numbers are uncountable
Question 5. Prove that square root of 2 is irrational
PROOFS:
1. State the prove The Density Theorem for Rational Numbers
2. Prove that irrational numbers are dense in the set of real numbers
3. Prove that rational numbers are countable
4. Prove that real numbers are uncountable
5. Prove that square root of 2 is irrational
Prove that the rational numbers do not satisfy the least upper
bound axiom. In particular, if a subset (S) of the rational numbers
is bounded above and M is the set of all rational upper bounds of
S, then M may not have a least element.
Let A be an infinite set and let B ⊆ A be a subset. Prove:
(a) Assume A has a denumerable subset, show that A is equivalent
to a proper subset of A.
(b) Show that if A is denumerable and B is infinite then B is
equivalent to A.
Prove the following:
(a) Let A be a ring and B be a field. Let f : A → B be a
surjective homomorphism from A to B. Then ker(f) is a maximal
ideal.
(b) If A/J is a field, then J is a maximal ideal.
Let A = Z and let a, b ∈ A. Prove if the following binary
operations are (i) commutative, (2) if they are associative and (3)
if they have an identity (if the operations has an identity, give
the identity or show that the operation has no identity).
(a) (3 points) f(a, b) = a + b + 1
(b) (3 points) f(a, b) = a(b + 1)
(c) (3 points) f(a, b) = x2 + xy + y2
1. Prove that given n + 1 natural numbers, there are always two
of them such that their difference is a multiple of n.
2. Prove that there is a natural number composed with the digits
0 and 5 and divisible by 2018.
both questions can be solved using pigeonhole principle.
Use the Well-Ordering Principle of the natural numbers to
prove that every positive
rational number x can be expressed as a fraction x = a/b where
a and b are postive
integers with no common factor.