Is the set of rational numbers countable?

Is the set of rational numbers countable?

The set of all rationals in [0, 1] is countable. Clearly, we can define a bijection from Q ∩ [0, 1] → N where each rational number is mapped to its index in the above set. Thus the set of all rational numbers in [0, 1] is countably infinite and thus countable.

Is the set of all points in the plane with rational coordinates countable?

The set of all points in the plane with rational coordinates. Solution: COUNTABLE. Proof: The given set is Q × Q. Since Q is countable and the cartesian product of finitely many countable sets is countable, Q × Q is countable.

How do you prove that a subset of a countable set is countable?

Every subset of a countable set is countable, proven by…

1. Lemma: Let f:A→B be a bijection, C⊆A, and f|C:C→B the restriction of f to C. Then f|C is a bijection.
2. Proof: Since f is a bijection, b=f(c) is uniquely defined for each c∈C, b∈B.
3. Since A is countable, there is a bijection ϕ:A→N.

Is countable union countable set countable?

Theorem: Every countable union of countable sets is countable. A set X is countable if and only if there exists a surjection f : N → X. Proof. If such a surjection exists, then X is countable by 7.3.

How do you tell if a set is countable or uncountable?

A set S is countable if there is a bijection f:N→S. An infinite set for which there is no such bijection is called uncountable. Every infinite set S contains a countable subset. Every infinite set S contains a countable subset.

What is countable and uncountable set?

The most concise definition is in terms of cardinality. A set S is countable if its cardinality |S| is less than or equal to (aleph-null), the cardinality of the set of natural numbers N. A set S is countably infinite if |S| = . A set is uncountable if it is not countable, i.e. its cardinality is greater than.

Is countable union of countable sets countable?

How is a countable set defined?

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

How do you prove that rational numbers are countable?

An easy proof that rational numbers are countable. A set is countable if you can count its elements. Of course if the set is finite, you can easily count its elements. If the set is infinite, being countable means that you are able to put the elements of the set in order just like natural numbers are in order.

How do you prove that a set is countable?

Here is a quick informal account of a standard proof that the set is countable: we can list its elements in the order , , , , , , , and so on. (This is informal because I have talked about “lists” and have not actually defined how the sequence continues.)

How do you prove that Q+ is countable?

Proof. Because Q+ contains the natural numbers, it is infinite, so we need only show it is countable. Define g: N×N→ Q+ by g(m,n) = m/n. Since every positive rational number can be written as a quotient of positive integers, g is surjective. Since N× Nis countable, it follows from Theorem 5(b) above that Q+ is countable. Exercise 1.

How do you prove that a function is countable?

A tool that is more often presented in treatments of countability is the fact that a countable union of countable sets is countable. That translates into the more general principle that if you can find a function such that each has at most countably many preimages, then is countable.