If aa and bb are countable sets, then so is a∪ba∪b.

Answer with Step-by-step explanation:

We are given that A and B are two countable sets

We have to show that if A and B are countable then is countable.

Countable means finite set or countably infinite.

Case 1: If A and B are two finite sets

Suppose A={1} and B={2}

={1,2}=Finite=Countable

Hence, is countable.

Case 2: If A finite and B is countably infinite

Suppose, A={1,2,3}

B=N={1,2,3,...}

Then, ={1,2,3,....}=N

Hence, is countable.

Case 3:If A is countably infinite and B is finite set.

Suppose , A=Z={..,-2,-1,0,1,2,....}

B={-2,-3}

=Z=Countable

Hence, countable.

Case 4:If A and B are both countably infinite sets.

Suppose A=N and B=Z

Then,==Z

Hence, is countable.

Therefore, if A and B are countable sets, then is also countable.

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step-by-step explanation:

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