• (c) These measures of discrimination satisfy MON. For the r-th identity group, rep, we define a vector у from x, where y] > x^{)r and y] > x] for all i Ф r, then y > x => l²¹(y) > ^(x), and for given A, у > x =>■ I²²(y) > /^(x).
• (d) The minimum values of these measures are equal to 0 when there is no unequal representation of the identity groups across the categories of wellbeing indicator. When there is under- or overrepresentation of the groups across the categories of the well-being indicator and this malrepresentation rises then the value of x?. -> oo, and l²¹ —> d (let), and I²² —» d (let), where d and d are countably infinite values.

The segregation measures f²¹ and l²² satisfy ABUC and AWUC. These proofs are same as the proof of Proposition 1.2. ■

Now we use another proposition to prove that the measures of segregation satisfy RUOCI.

Proposition 10.7. The segregation measures I²¹ and I²² satisfy RUOCI.

Proof: In the case of RUOCI, regrouping of the population occurs and one group may be decomposed into two or more groups and/or two or more identity groups may be combined into one. We define у from x by A* = X] + X^!(, and Af = Ajf for all i ф r, for x, у € R, г, и e p_x and t, v e py.

It can be proved that,

And finally we have,/²¹ (x?) =/²¹(y°) and/²²(x?.) =/²²(yj|)

The proof is similar to the case of combining groups by A;( = A) + A* and Af = Af for i Ф r,l.

The segregation measures l²¹ and I²² satisfy PSI, but the segregation measure f²¹ only satisfies PCI. These proofs are same as the proofs of the Proposition 1.4. ■

Proposition 10.8. The segregation measures f¹¹ and I^й satisfy TP and EP.

Proof: Transfer principle states that, when initially > Ki or (~f)^X > (/)*/ and we define у from x, where x,y € R+, r 6 y>, s, l e , by Х-ф = A* — p = (~f)^x ~ P and A^y, = X^х, + p = {-f)^x + p, where p = ^ e (0,1), and A*- = T for all i Ф r and j Ф s, l, then there is a decline in the value of segregation measured by the specific segregation measures that satisfy this property

Through the transfer of a person, the value of x% declines and x°_rl rises, but the sum total of the value of (x® + x°_rl) < (y% + y°_rl), since p =fy > 0.

As a result,/¹ (xg) >/¹(y°) and/²(xg) >/²(yg).

Ч^ГЮ-/Ю1 = о-

The proof for I³² is similar to the proof of f³¹.

(b) = { J2f=i ^where xfeBlK-V and В e R+.

If 3c^° -> 1 for all i e

The methods involved in the proofs to show that the measures f³¹ and I³² satisfy ABUC, AWUC, PCI and PSI and are identical to the methods involved in Appendices I and II. We use the following Proposition 3.2 to show that the measures I³¹ and I³² satisfy RUOCI. ■

Proposition 10.10. The segregation measures I^3] and I³² satisfy RUOCI.

Proof: The segregation measures l³¹ and I³² having the following forms:

Or,

Now, RUOCI implies that for all x, у e IR, г, и e p_x and t,v e y>y, we define у from x by A* = X) + A^ and A* = Af for all i Ф r. For A* = X) + A?, we have Kj ⁼ tfj + fyy for all j e ф. Therefore, after the subdivision of the r-th group into f-th and i'-th groups we have the cumulative population distributions of these groups as: Г*. = Г-JJ. + Г-T for all j e ф, though we have £(Pfy = Е(Гр.

Thus, the value of щА- is invariant with this subdivision of one group and x®s are invariant with this subdivision of population of the particular group.

The proof of the second part of this property is almost similar to the first part. Now Proposition 3.3 is used to show that the segregation measures I³¹ and I³² satisfy SIC. ■

Proposition 10.11. The segregation measures I³¹ and 1³² satisfy SIC.

Proof: This property states that for all x,y € R, г, и e for x = (x,X2,..., x_r, .,x_M) and у = (yi,y₂,... ,y_u,.. .,y„... ,ум) => /(x) = /(y). However, for all Xj, yi € R and s,l e ф for Xj = (3c,i,xq, ■ ■ ■,Xj_S,...,3c,7,...,x,k) and