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1.Relation and Function
easy
Let $R$ be the relation on the set $R$ of all real numbers defined by $a \ R \ b$ if $|a - b| \le 1$. Then $R$ is
A
Reflexive and Symmetric
B
Symmetric only
C
Transitive only
D
Anti-symmetric only
Solution
(a) $|a – a| = 0 < 1$
$\therefore$ $R$ is reflexive.
Again $a\ R\ b$ $⇒$ $|a – b| \le 1 \Rightarrow |b – a| \le 1 \Rightarrow bRa$
$\therefore$ $R$ is symmetric, Again $1R{1 \over 2}$ and ${1 \over 2}R1$ but ${1 \over 2} \ne 1$
$\therefore$ $R$ is not anti-symmetric.
Further, $1\ R\ 2$ and $2\ R\ 3$ but $1\, R\,3$, [$\because \,|1 – 3| = 2 > 1$]
$\therefore$ $R$ is not transitive.
Standard 12
Mathematics
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