Question

To verify that for two sets A and B , n(AxB) = pq and the total number of relation from A to B is 2 raise to power pq where n(A) = p and n(B) = q

Answer 1

SOLUTION

TO VERIFY

1. For two sets A and B , n(A × B) = pq

2. The total number of relation from A to B is

$\sf{ {2}^{pq} }$where n(A) = p and n(B) = q

EVALUATION

1. Let A = { a ,b } & B = { 1,2}

∴ A × B = {(a, 1), (a,2), (b, 1), (b,2) }

∴ n(A) = 2 , n(B) = 2 & n(A×B) = 4

∴ n(A×B) = n(A) × n(B)

Hence verified

2. Let A = { a } & B = { 1,2}

∴ A × B = {(a, 1), (a,2) }

Then relations from A to B are

{(a, 1), (a,2) }, {(a, 1)}, {(a,2) }, $\Phi$

So the total number of relations

= 4

$= {2}^{2}$

$= {2}^{1 \times 2}$

Hence verified

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Answer 2

$\huge\underline{\bf \orange{AnSweR :}}$

1. Let A = { a ,b } & B = { 1,2}

∴ A × B = {(a, 1), (a,2), (b, 1), (b,2) }

∴ n(A) = 2 , n(B) = 2 & n(A×B) = 4

∴ n(A×B) = n(A) × n(B)

Hence verified.

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2. Let A = { a } & B = { 1,2}

∴ A × B = {(a, 1), (a,2) }

Then relations from A to B are, {(a, 1), (a,2) }, {(a, 1)}, {(a,2) }, Φ

So the total number of relations :

→ 4

→ 2²

→ 1×2²

Hence verified.