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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 (52.2k points)
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by Loyal (5.7k 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,, 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 (7.8k points)
+1 vote

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 (4.5k 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 (1.6k points)
–3 votes
It is TRUE.
by Loyal (7.2k points)
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