The relation R in the set of real numbers, defined as R = {(a, b): 1 + ab > 0}, is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Reflexive and symmetric but not transitive
Reflexive: for every real a, \( 1+a\cdot a=1+a^2\gt 0 \), so \( (a,a)\in R \).
Symmetric: if \( 1+ab\gt 0 \), then \( 1+ba=1+ab\gt 0 \), so \( (a,b)\in R\Rightarrow(b,a)\in R \).
Not transitive: a counter-example is enough, with a = −1, b = 0, c = 2:
\( \displaystyle (-1,0):\ 1+0=1\gt 0,\ \text{so } (-1,0)\in R \)
\( \displaystyle (0,2):\ 1+0=1\gt 0,\ \text{so } (0,2)\in R \)
\( \displaystyle (-1,2):\ 1-2=-1\lt 0,\ \text{so } (-1,2)\notin R \)
So R is reflexive and symmetric but not transitive.