Let R be the relation on the set R of all real numbers defined by a R b if and only if $|a-b|≤1$. Then R is ___________.

Symmetric

$|a-a|=a<1$

Therefore, $a\,R\,a\,∀\,a∈R$

Therefore, R is reflexive.

Again a R b, $|a-b|≤1⇒|b-a|≤1⇒b\,R\,a$

Therefore, R is symmetric.

Again $1\,R[\frac{1}{2}]$ and $[\frac{1}{2}]R\,1$ but $[\frac{1}{2}]≠1$

Therefore, R is not anti-symmetric.

Further, 1 R 2 and 2 R 3 but $[\frac{1}{R3}]$, [Because, |1 - 3| = 2 > 1]

Hence, R is not transitive.