For two sets A and B define the Cartesian Product as A times

For two sets A and B. define the Cartesian Product as A times B = {(a, b): a elementof A, b elementof B}. Given two countable sets A and B, prove that A times B is countable. Given a finite number of countable sets A_1, A_2, ellipsis, A_n, prove that A_1 times A_2 times ellipsis times A_n is countable. Consider an infinite number of countable sets: B_1, B_2, ellipsis Under what condition(s) is B_1 times B_2 times ellipsis countable? Prove that if this condition is violated, B_1 times B_2 times ellipsis is uncountable.

Solution

Ans(a):

Given that A and B are countable sets then say there are m and n elements in A and B respectively.

in that case AxB will have total mxn elements. {by property of cartesian product}

we know that product of two countable numbers m and n is also countable hence mxn is also countable.

Which means there are finite elements in AxB.

Hence AxB is countable.

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Ans(b):

Given that A1,A2,...,An are countable sets then say m1,m2,...,mn are number of elements in those sets respectively.

in that case A1xA2x...xAn will have total m1xm2x...xmn elements. {by property of cartesian product}

we know that product of countable numbers is also countable hence m1xm2x...xmn is also countable.

Which means there are finite elements in A1xA2x...xAn.

Hence A1xA2x...xAn is countable.