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Question
If * is defined on the set R of all real numbers by *: a*b = `sqrt(a^2 + b^2 ) `, find the identity elements, if it exists in R with respect to * .
Solution
Let * be an operation defined on R
a * b = `sqrt(a^2 + b^2 ) `
Let ‘e’ be the identity element
then a*e = e*a = a
`sqrt(a^2 + e^2 ) = sqrt(e^2 + a^2 ) = a `
B squaring on both side
e2 + a2 = a2
e2 = 0 ⇒ e = 0
a*0 = `sqrt( a^2 + 0^2 ) ` = |a| ≠ a
i.e. if a = -2 (or any other negative real no.)
then a* 0 ≠ -2
since a*0 = `sqrt(a^2) `= | a| = 2 ≠ - 2
Hence identity element does not exists.
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