अशून्य सम्मिश्र संख्याओं ' C ' के समुच्चय पर परिभाषित सम्बन्ध `R` इस प्रकार है कि `Z_(1) R Z_(2) hArr (Z_(1)-Z_(2))/(Z_(1)+Z_(2))` एक वास्तविक संख्या है । ज्ञात कीजिए कि `R-`

Options

A. स्वतुल्य तथा सममित परन्तु संक्रामक नहीं
B. सममित तथा संक्रामक परन्तु स्वतुल्य नहीं
C. संक्रामक परन्तु ना तो स्वतुल्य ना ही सममित
D. स्वतुल्य , सममित, संक्रामक

Answer

D

Explanation

`(i) Z in C` के लिये
`._(Z)R_(Z)rArr(Z-Z)/(Z+Z)` एक वास्तविक संख्या है ।
`rArr 0` एक वास्तविक संख्या है । जो सत्य है ।
अतः R स्वतुल्य है ।
(ii) माना `Z_(1),Z_(2)inC` और `._(Z_(1))R_(Z_(2))`
अब `._(Z_(1))R_(Z_(2))rArr(Z_(1)-Z_(2))/(Z_(1)+Z_(2))` एक वास्तविक संख्या है ।
`rArr-((Z_(2)-Z_(1))/(Z_(2)+Z_(1)))` एक वास्तविक संख्या है ।
`rArr-(Z_(2)-Z_(1))/(Z_(2)+Z_(1))` एक वास्तविक संख्या है ।
`rArr._(Z_(2))R_(Z_(1))`
अतः R सममित है ।
(iii) माना `Z_(1),Z_(2),Z_(3) in C` और `._(Z_(1))R_(Z_(2))` तथा `._(Z_(2))R_(Z_(3))`
अब `._(Z_(1))R_(Z_(2))rArr(Z_(1)-Z_(2))/(Z_(1)+Z_(2))` एक वास्तविक संख्या है ।
`rArr((x_(1)+iy_(1))-(x_(2)+iy_(2)))/((x_(1)+iy_(1))+(x_(2)+iy_(2)))` एक वास्तविक संख्या है ।
जहाँ `Z_(1)=x_(1)+iy_(1)` और `Z_(2)=x_(2)+iy_(2)`
`rArr((x_(1)-x_(2))+i(y_(1)-y_(2)))/((x_(1)+x_(2))+i(y_(1)+y_(2)))` एक वास्तविक संख्या है ।
`rArr([(x_(1)-x_(2))+i(y_(1)-y_(2))][(x_(1)+x_(2))-i(y_(1)+y_(2))])/([(x_(1)+x_(2))+i(y_(1)+y_(2))][(x_(1)+x_(2))-i(y_(1)+y_(2))])` एक वास्तविक संख्या है ।
`rArr((x_(1)^(2)-x_(2)^(2))+(y_(1)^(2)-y_(2)^(2))+i{(x_(1)+x_(2))(y_(1)-y_(2))-(x_(1)-x_(2))(y_(1)+y_(2))})/((x_(1)+x_(2))^(2)+(y_(1)+y_(2))^(2))` एक वास्तविक संख्या है । `rArr i` का गुणांक `=0`
`rArr(x_(1)+x_(2))(y_(1)-y_(2))-(x_(1)-x_(2))(y_(1)+y_(2))=0`
`rArr x_(2)y_(1)-x_(1)y_(2)=0`
`rArr (x_(1))/(y_(1))=(x_(2))/(y_(2))`
`:. ._(Z_(1))R_(Z_(2))rArr(x_(1))/(y_(1))=(x_(2))/(y_(2))`
इसी प्रकार `._(Z_(2))R_(Z_(3))rArr(x_(2))/(y_(2))=(x_(3))/(y_(3))`
अतः `._(Z_(1))R_(Z_(2))` और `._(Z_(2))R_(Z_(3))`
`rArr (x_(1))/(y_(1))=(x_(2))/(y_(2))`और `(x_(2))/(y_(2))=(x_(3))/(y_(3))`
`rArr(x_(1))/(y_(1))=(x_(3))/(y_(3))rArr ._(Z_(1))R_(Z_(3))`
अतः R सम्बन्ध संक्रमक है ।
इस प्रकार दिया सम्बन्ध R स्वतुल्य, सममित और संक्रमक है ।
अर्थात R एक तुल्यता सम्बन्ध है । `" "` यही सिद्ध करना था ।