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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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Suppose A 1 , A 2 , A 3 , .... is an infinite sequence of countable set. Recall that
i = 1 A i = { x | x A i } for some positive integer i.}.
Prove that i = 1 A i is countable. (In other words, prove that a countably infinite union of countable set is countable.)

To determine

To prove:

i=1Ai is countable, where A1,A2,A3,..... is an infinite sequence of countable sets.

Explanation

Given information:

A1,A2,A3,..... is an infinite sequence of countable sets.

i=1Ai={x|xAi for some positive integer i}

Concept used:

A function is said to be one-to-one function if the distinct elements in domain must be mapped with distinct elements in co-domain.

A function is onto function if each element in co-domain is mapped with atleast one element in domain.

Countable function should be one-to-one and onto.

Proof:

Consider A1,A2,A3,..... is an infinite sequence of countable sets such that

A1={a11,a12,a13,......}A2={a21,a22,a23,......}A3={a31,a32,a33,......}

And so on.

Also i=1Ai={x|xAi for some positive integer i}

The object is to prove i=1Ai is countable.

Now, let us take the union of all the elements of A1,A2,A3,.....

i=1Ai=A1A2A3

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