The relation R defined in A = {1, 2, 3} by aRb if a2−b2≤5. Which of the following is false

Let a=1 ∴a2−b2l≤5⇒1−b2∣≤5⇒∣b2−1I≤5⇒b=1,2 Let a=2∴a2−b2≤5⇒4−b2∣≤5 Let 3∴a2−b2≤5⇒∣9−b2≤5⇒b2−9≤5⇒b=2,3∴R={(1,1),(1,2),(2,1),(2,2)(2,3),(3,2)(3,3)}R−1={(y,x):(x,y)∈R}⇒R={(1,1),(2,1),(1,2),(2,2)(3,2),(2,3)(3,3)}Domain of R={x:(x,y)∈R}={1,2,3}⇒Range of R={y:(x,y)∈R}={1,2,3}

The relation R defined in A = {1, 2, 3} by aRb if a2−b2≤5. Which of the following is false

Let a=1 ∴a2−b2l≤5⇒1−b2∣≤5⇒∣b2−1I≤5⇒b=1,2 Let a=2∴a2−b2≤5⇒4−b2∣≤5 Let 3∴a2−b2≤5⇒∣9−b2≤5⇒b2−9≤5⇒b=2,3∴R={(1,1),(1,2),(2,1),(2,2)(2,3),(3,2)(3,3)}R−1={(y,x):(x,y)∈R}⇒R={(1,1),(2,1),(1,2),(2,2)(3,2),(2,3)(3,3)}Domain of R={x:(x,y)∈R}={1,2,3}⇒Range of R={y:(x,y)∈R}={1,2,3}

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The relation R defined in A = {1, 2, 3} by aRb if $|a^{2} - b^{2}| \leq 5$ . Which of the following is false

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R = {(1, 1), (2, 2), (3, 3), (2, 1), (1, 2), (2, 3), (3, 2)}

$R^{- 1} = R$

Domain of R = {1, 2, 3}

Range of R = {5}

answer is D.

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Detailed Solution

Let $a = 1 ∴ a^{2} - b^{2} l \leq 5 \Rightarrow 1 - b^{2} ∣\leq 5$
$\begin{array}{l} \Rightarrow∣ b^{2} - 1 I \leq 5 \Rightarrow b = 1, 2 \\ Let a = 2 ∴ |a^{2} - b^{2} \leq 5 \Rightarrow| 4 - b^{2} ∣\leq 5 \\ Let 3 ∴ |a^{2} - b^{2}| \leq 5 \Rightarrow∣ 9 - b^{2} \leq 5 \\ \Rightarrow |b^{2} - 9| \leq 5 \Rightarrow b = 2, 3 \\ ∴ R = {(1, 1), (1, 2), (2, 1), (2, 2) (2, 3), (3, 2) (3, 3)} \\ R^{- 1} = {(y, x) : (x, y) \in R} \\ \Rightarrow R = {(1, 1), (2, 1), (1, 2), (2, 2) (3, 2), (2, 3) (3, 3)} \end{array}$
Domain of $R = {x : (x, y) \in R} = {1, 2, 3}$
$\Rightarrow$ Range of $R = {y : (x, y) \in R} = {1, 2, 3}$