Let . Then the minimum number of elements, required to be added in R to make it a transitive relation, is ________. [2026]

Question

Let $R = {(x, y) \in N \times N : \log_{e} (x + y) \leq 2}$ . Then the minimum number of elements, required to be added in R to make it a transitive relation, is ________. [2026]

Smart Achievers · Accepted Answer

(15)

Given : $\log_{e} (x + y) \leq 2$

$\Rightarrow x + y \leq e^{2}$

Since, $e^{2} \approx 7.29, so x + y \leq 7$

$∴$ R = {(1, 1), (1, 2), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (3, 1) (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (6, 1)}

For R to be transitive, we have to add (6, 2), (6, 3), (6, 4), (6, 5), (6, 6), (5, 3), (5, 4), (5, 5), (5, 6), (4, 4), (4,5), (4, 6), (3, 5),(3, 6), (2, 3) i.e., 15 elements

Solution

Q.
Let $R = {(x, y) \in N \times N : \log_{e} (x + y) \leq 2}$ . Then the minimum number of elements, required to be added in R to make it a transitive relation, is ________. [2026]

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Solution

Q. Let R={(x,y)∈N×N:loge(x+y)≤2}. Then the minimum number of elements, required to be added in R to make it a transitive relation, is ________. [2026]

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Q.
Let $R = {(x, y) \in N \times N : \log_{e} (x + y) \leq 2}$ . Then the minimum number of elements, required to be added in R to make it a transitive relation, is ________. [2026]