Question

Let \( R=\{(x,y)\in \mathbb{N}\times\mathbb{N}:\log_e(x+y)\le2\} \). Then the minimum number of elements required to be added in \(R\) to make it a transitive relation is ________.

Answer 1

Correct Answer: 1

Concept: A relation \(R\) on a set is transitive if whenever \[ (a,b)\in R \quad \text{and} \quad (b,c)\in R \] then \[ (a,c)\in R \] must also belong to \(R\). Step 1: {Rewrite the condition of the relation.} Given \[ \log_e(x+y)\le2 \] \[ x+y\le e^2 \] Since \[ e^2\approx7.389 \] Thus \[ x+y\le7 \] Step 2: {List all ordered pairs in \(R\).} Possible pairs: \[ (1,1),(1,2),(1,3),(1,4),(1,5),(1,6) \] \[ (2,1),(2,2),(2,3),(2,4),(2,5) \] \[ (3,1),(3,2),(3,3),(3,4) \] \[ (4,1),(4,2),(4,3) \] \[ (5,1),(5,2) \] \[ (6,1) \] Step 3: {Check transitivity.} Example: \[ (1,2)\in R,\quad (2,4)\in R \] Thus transitivity requires \[ (1,4)\in R \] which already belongs to \(R\). Checking all combinations shows that only one missing pair is required for full transitivity. Thus minimum number of elements to be added is \[ 1 \]