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RELATIONS AND FUNCTIONS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSRELATIONS AND FUNCTIONS3 Marks
Question
For the binary operation $\times _7$ on the set $S = \{1, 2, 3, 4, 5, 6\},$ compute $3^{−1} \times _7 4.$
✓

Answer

Finding identity element:
Here,
$1\times _71 =$ Remainder obtained by dividing $1 \times 1$ by $7 = 1$
$3\times _74 =$ Remainder obtained by dividing $3 \times 4$ by $7 = 5$
$4\times _75 =$ Remainder obtained by dividing $4 \times 5$ by $7 = 6$
So, the composition table is as follows:
$\times _7$ $1$ $2$ $3$ $4$ $5$ $6$
$1$ $1$ $2$ $3$ $4$ $5$ $6$
$2$ $2$ $4$ $6$ $1$ $3$ $5$
$3$ $3$ $6$ $2$ $5$ $1$ $4$
$4$ $4$ $1$ $5$ $2$ $6$ $3$
$5$ $5$ $3$ $1$ $6$ $4$ $2$
$6$ $6$ $5$ $4$ $3$ $2$ $1$
We observe that all the elements of the first row of the composition table are same as the top$-$most row.
So, the identity element is $1.$
Also, $3\times _{7 }5 = 1$
So, $3^{-1} = 5$
Now,
$3^{-1}\times _7 4 $
​​​​​​​$= 5\times _7 4 $
$= 6$

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