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Binary Operations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsBinary Operations3 Marks
Question
Let $+_6 ($addition modulo $6)$ be a binary operation on $S =\{0, 1, 2, 3, 4, 5\}.$ Write the value of $2 + _64^{−1}+ _63^{−1}.$
✓

Answer

The composition table for $+_6$ on the set $S = \{0, 1, 2, 3, 4, 5\}$ is
$+_6$ $0$ $1$ $2$ $3$ $4$ $5$
$0$ $0$ $1$ $2$ $3$ $4$ $5$
$1$ $1$ $2$ $3$ $4$ $5$ $0$
$2$ $2$ $3$ $4$ $5$ $0$ $1$
$3$ $3$ $4$ $5$ $0$ $1$ $2$
$4$ $4$ $5$ $0$ $1$ $2$ $3$
$5$ $5$ $0$ $1$ $2$ $3$ $4$
$'0'$ is the identity element for $+_6$ from the table it is clear that
$4^{-1} = 2$ and $3^{-1} = 3$
Now, $2 + _64^{-1} + _63^{-1} = 2 + _62 + _63$
$= 4 + _63$
$= 1$

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