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Purely Real

Let A = x : x...

Question

Let $A = {x : x = 2 n + 1, n \in W}$ and $B = {y : y = 2 n, n \in N} .$ If a relation $R$ is defined from $A$ to $B,$ then which of the following is void relation?

R = {(x, y) : x + y = 11}

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R = {(x, y) : x + 2 y = 21}

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R = {(x, y) : 2 x + y = 14}

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R = {(x, y) : x + y = 8}

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Solution

The correct option is D $R = {(x, y) : x + y = 8}$
$A = {x : x = 2 n + 1, n \in W}$
$\Rightarrow A = {1, 3, 5, 7, 9, . . .}$
$B = {y : y = 2 n, n \in N}$
$\Rightarrow B = {2, 4, 6, 8, 10, . . .}$

$R = {(x, y) : x + y = 11}$
Here, $R = {(1, 10), (3, 8), (5, 6), (7, 4), (9, 2)}$
It is not a void relation.

$R = {(x, y) : x + 2 y = 21}$
Here, $R = {(1, 10), (5, 8), (9, 6), (13, 4), (17, 2)}$
It is not a void relation.

$R = {(x, y) : 2 x + y = 14}$
Here, $R = {(1, 12), (3, 8), (5, 4)}$
It is not a void relation.

$R = {(x, y) : x + y = 8}$
Since, $x$ is odd number and $y$ is even number, so sum of $x + y$ can never be even number.
Hence, it is a void relation.

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