A relation R on the set of complex numbers is defined by z1 Rz2 if and only if (z1−z2)/(z1+z2) is real. Show that R is an equivalence relation.

Question

A relation R on the set of complex numbers is defined by z1​ Rz2​ if and only if (z1​−z2​)/(z1​+z2​) is real. Show that R is an equivalence relation.

Filo · Accepted Answer

Here z1​ R z2​ for all complex number z1​.For we have z1​+z2​z1​−z1​​=0 is real, then z1​+z2​z1​−z1​​ is real, then z2​+z1​z2​−z1​​ is also real.Hence z1​Rz2​→z2​Rz1​.We now show that R is transitive.Let z1​=x1​+iy1​,z2​=x2​+iy2​ and z3​=x3​+iy3​ be three complex numbers such that z1​Rz2​ and z2​Rz3​. Now z1​Rz2​⇒z1​+z2​z1​−z2​​ is real ⇒(x1​+x2​)+i(y1​+y2​)(x1​−x2​)+i(y1​−y2​)​ is real⇒(x1​+x2​)2+(y+y2​)2][(x1​−x2​)+i(y1​−y2​)][(x1​+x2​)−i(y1​−y2​)]​ is real⇒(y1​−y2​)(x1​+x2​)−(x1​−x2​)−(y1​+y2​)=0⇒2x2​y1​−2y2​x1​=0⇒(x1​/y1​)=(x2​/y2​) ........(1)Similarlly z2​Rz3​=y2​x2​​=y3​x3​​       ...(2)From (1) and (2), we have y1​x1​​=y3​x3​​Hence z1​Rz3​.Thus z1​Rz2​ and z2​Rz3​⇒z1​Rz3​.Hence R is transitive. It follows that R is an equivalence relation.Note: If we consider R is a relation on the set of all complex number but 0(~R)0 for we have 0+00−0​=00​ which is indeterminate. Hence in order that R may be an equivalence relation the set on which R is defined must be non-zero complex numbers.

Question Text	A relation R on the set of complex numbers is defined by $z_{1}$ R $z_{2}$ if and only if $(z_{1} - z_{2}) / (z_{1} + z_{2})$ is real. Show that R is an equivalence relation.
Answer Type	Text solution:1
Upvotes	150

A relation R on the set of complex numbers is defined by $z_{1}$ R $z_{2}$ if and only if $(z_{1} - z_{2}) / (z_{1} + z_{2})$ is real. Show that R is an equivalence relation.

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A relation R on the set of complex numbers is defined by z1​ Rz2​ if and only if (z1​−z2​)/(z1​+z2​) is real. Show that R is an equivalence relation.

Text solutionVerified

Practice questions from similar books

A relation R on the set of complex numbers is defined by $z_{1}$ R $z_{2}$ if and only if $(z_{1} - z_{2}) / (z_{1} + z_{2})$ is real. Show that R is an equivalence relation.