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Every subset of a countable set is countable.

State whether the above statement is true or false with reason.
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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

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} ---(Countable)------------ cardinality =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)

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It is TRUE.