Question

If AA and BB are countable sets, then so is A∪BA∪B.

Answer 1

To remedy confusions like yours and to avoid the needless case analyses, I prefer to define X to be countable if there is a surjection from N to X.

This definition is equivalent to a few of the many definitions of countability, so we are not losing any generality.

It is a matter of convention whether we allow finite sets to be countable or not (though, amusingly, finite sets are the only ones whose elements you could ever finish counting).

So, if A and B be countable, let f:N→A and g:N→B be surjections. Then the two sequences (f(n):n⩾1)=(f(1),f(2),f(3),…) and (g(n):n⩾1)=(g(1),g(2),g(3),…) eventually cover all of A and B, respectively; we can interleave them to create a sequence that will surely cover A∪B:

(h(n):n⩾1):=(f(1),g(1),f(2),g(2),f(3),g(3),…).

An explicit formula for h is h(n)=f((n+1)/2) if n is odd, and h(n)=g(n/2) if n is even.

Hope it helps uh mate...✌

Answer 2

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
`A\cup B` 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}

`A\cup B`={1,2}=Finite=Countable

Hence,
`A\cup B` is countable.

Case 2: If A finite and B is countably infinite

Suppose, A={1,2,3}

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

Then,
`A\cup B`={1,2,3,....}=N

Hence,
`A\cup B` 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}

`A\cup B`=Z=Countable

Hence,
`A\cup B` countable.

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

Suppose A=N and B=Z

Then,
`A\cup B`=
`N\cup Z`=Z

Hence,
`A\cup B` is countable.

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