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Find the whichof theoperations given abovehas identity ? |
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Answer» Solution :(i) Q , a*B= a-b Leta* e = a e*a `rArr""a- e = a= e - a ` Now, `""a-e=a rArre=0` and `"" e-a = a rArr e =2a` butthe identity is alwaysunique. `THEREFORE` In Q the identity elements does notexist with respect to the operationa*b = a-b (ii) In Q,`"a * b" = a^(2) +b^(2)` Leta* e = a * a `rArr""a^(2) + e^(2) =a = e^(2) + a^(2)` If a=-2 then e does not exist. `therefore` In Q, the identityelements does not exist with respect to the operate a* b `= a^(2) + b^(2)` (iii)In Q, a *b `= a^(2) + b^(2)` Let a * e = a = e * a `rArr` a + ae = a=e + ea Now , a + ae = a `rArr` ae = 0 `rArr` e = 0 ande + ea = `rArr e =(a)/(1+a), a NE 0` but the identify element is always unique `therefore` In Q, the identity element does not exist with respect to the OPERATION a * b = a + ab. (IV) InQ, `""a * b = (a-b)^(2)` `rArr""(a -e)^(2) a = (e-a)^(2)` For `"'" a=-3` `""(-3-e)^(2) = - 3`Whichis not possible. `therefore` In Q the identity element does not exist with respect to the operation a * b `= (-b)^(2)` (v) In Q, `a** b = (ab)/(4)` Let a* e= a = e*a. `rArr(ae)/(4) = a= (ea)/(4)` `rArre =4` `therefore` Operation`a ** b = (ab)/(4)` (vi) In Q , `a*b = ab^(2)` Let a*e = a = e*a `rArr ""ae^(2) =a = ea^(2)` Now `""ae^(2) = a "" rArr"" e = pm 1` `and "" a=ea^(2) "" rArr ""e = (1)/(a)` but the identity element is always unique. `therefore` In Q ·the identity element does not exist with respect to the operation `a ** b = ab^(2)` |
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