The Gateway to Computer Science Excellence
First time here? Checkout the FAQ!
x
+11 votes
556 views
Every subset of a countable set is countable.

State whether the above statement is true or false with reason.
in Set Theory & Algebra by Veteran (52.1k points)
retagged by | 556 views

5 Answers

+10 votes
Best answer
by Loyal (5.6k points)
edited by
+3 votes

Theorem . Every subset of a countable set is countable.

Proof.  Suppose a1,a2,a3,....... is an enumeration of the countable set A and B is any nonempty subset of A. If, for some n∈ N, the element 'an' (a subscript n) belongs to B, then we assign the natural number n to it. For each n∈ N let k(n) denote the number of elements among a1,a2,a3,a4,...an, which belong to the subset B. Then ,0≤ k(n) ≤n . Therefore, B is countable by the Countability Lemma.

Every subset of a countable set is countable.  TRUE

by Loyal (7.5k points)
0 votes

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set.

R = {1,2,3,4............................} 

R1= {1,2}--------(Countable))------------ cardinality =2

R2={2,3}-------(Countable))------------ cardinality =2

R3={3,4}-------(Countable))------------ cardinality =2

so Every subset of a countable set is countable (true)

by Active (4.2k points)
edited by
–1 vote
Every subset of countable set is countable ,hence answer is true.

We should also remember that ,

An infinite subset of countable set is also countable.

Hope it helps
by Active (1.3k points)
–3 votes
It is TRUE.
by Loyal (7.1k points)

Related questions

Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
49,845 questions
54,783 answers
189,421 comments
80,415 users