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THE INHERITANCE OF LEGAL HEIRS: MATHEMATICAL PRESENTATION KUNLE OLADEJI BABALOLA (ABUU ABDILQUDDUUS AS-SALAFEE) Abstract. “ BismiLlah, wasalaatu wasalaam ’alaa rasuuliLlah, wa ’alaa aalihi wasahabihi wa man waalaah ”. The object of this paper is the mathematical presentation of the inheritance of legal heirs of a deceased Muslim’s estate. The presentation, based on relevant shari’ah texts, depends largely on basic set theory approach with short representation equations to fit, alhamduliLlah . The shar’ah texts, however, are not stated in the paper, which is a quick indication that the paper is not self-sufficient. However, it will be resourceful in facilitating and enhanc- ing the understanding of the subject, especially for the mathematically- inclined students of knowledge, in shaa - a Allah . 1. Abbreviations The following abbreviations are used in the sequel. 1.0. D – The deceased 1.1. al - Usuul wal Furuu ′ (Roots and Branches) f – father, m – mother, s ( s ) – son(s), d ( s ) – daughter(s), gf – grandfa- ther, gm ( s ) – grandmother(s) [mother(s) of m , f and gf ], gs ( s ) – grand- son(s), gd ( s ) – granddaughter(s) through son(s), gsd – grandson(s) and granddaughter(s). 1.2. al - Hawaashee (Footnotes) fb ( s ) – full brother(s), fs ( s ) – full sister(s), cb ( s ) – consanguine brother(s), cs ( s ) – consanguine sister(s), csb – consanguine brother(s) and sister(s), fu – full uncle, hu – half uncle, ubs ( s ) – uterine brother(s) and sister(s), s 1 , s 2 , s 3 , s 4 – sons of fb, cb, fu, hu respectively. 1.3. az - Zawj (Spouses) p – spouse ( h – husband, w ( s ) – wives). 1.4. Mu ′ tiq q – mutiq or mutiqah and their ’asabatu bin nafs . 1
2 ABUU ABDILQUDDUUS AS-SALAFEE 2. Set Definitions Let L be the set of all legal heirs. Then: L = { s, d, gs, gd, f, gf, fb, fs, cb, cs, s 1 , s 2 , fu, hu, s 3 , s 4 , q, m, gm ( s ) , p, ubs } . Noting that gm ( s ) counts 3 while q , p and ubs each counts 2, the cardinality of L is 26. Each x ∈ L inherits either: (i) a fixed share ( fard ) or (ii) the residue (by ta ′ asiib ). Similarly, apart from exclusion by ascendancy or descendancy, each is either excludable by others or non-excludable by any. Based on this we define the following set. 2.1. Non-excludable Heirs A = { f, [ gf ] , m, [ gm ( s )] , s, [ gs, gd ] , d, p } The set, A , consists of the usuul , the furuu ′ u and the zawj . Except for proximity exclusions (that is nearer ascendants and descendants exclude the farther), no member of this set is excluded by any heir. The brackets [] means exclusion by nearer heirs to the deceased only. Any other heir not in A is excludable by one or more other heirs. 2.2. As - haabul furuudh F = { f, gf, m, gm ( s ) , h, w ( s ) , d ( s ) , gd ( s ) , fs ( s ) , cs ( s ) , ubs ( s ) } The set, F , consists all heirs who inherit fard (specified fraction). 2.3. al - ′ Asabatu bin nafs T 1 = { s, gs, f, gf, fb, cb, s 1 , s 2 , fu, hu, s 3 , s 4 } This set consists of residuaries (all heirs who inherit residue) by self. 2.4. al - ′ Asabatu bil ghayr T 2 = { d, gd, fs, cs } This set consists of residuaries by others. In the presence of their male counterparts, they inherit the residue together with them in ratio 2:1 ( lidh dhakar mithlu hadhil unthayayn ). 2.5. al - ′ Asabatu ma ′ al ghayr T 3 = { fs, cs } This is the set of residuaries with others. In the absence of their male counterparts, they take their places in T 1 as residuaries, if d or gd is present. 2.6. al - ′ Asabatu bis sabab T 4 = { q } This is the set of residuaries by circumstance.
INHERITANCE: MATHEMATICAL PRESENTATION 3 2.7. All ′ Asabah T = ∪ T i = { s, [ d ] , gs, [ gd ] , f, gf, fb, [ fs ] , cb, [ cs ] , s 1 , s 2 , fu, hu, s 3 , s 4 , q } This is the set of all residuaries. This set is ordered - higher residuary takes the residue. The brackets [] means ‘residuary by’. 2.8. Pure As - haabul furuudh T c = { m, [ gm ( s )] , p, ubs } This set consists of non-residuaries (those who inherit fard only). 2.9. Pure ′ Asabah T 1 − G = { s, gs, fb, cb, s 1 , s 2 , fu, hu, s 3 , s 4 } This set consists of residuaries only (those who inherit only the residue). This set is also ordered higher residuaries take residue. 2.10. The Intersection G = T ∩ F = { f, gf, d ( s ) , gd ( s ) , fs ( s ) , cs ( s ) } This is the set of those who inherit both the fard and the residue. 2.11. al - Hawaashee H = { fb, [ fs ] , cb, [ cs ] , s 1 , s 2 , fu, hu, s 3 , s 4 } This set consists of brothers, sisters and uncles of the deceased through his father, and the sons of the male ones among them. The set is also ordered - each higher member excludes the lower ones. The female members do so only by ta ′ asiib with d ( s ) or gd ( s ) of the deceased. This is indicated by []. 3. Six Golden exclusion Rules Rule 3.1. Nearer ascendant, descendant excludes farther ( f excludes gf ,...; m excludes gm ( s ),...; s excludes gs ( s ) , gd ( s ), ..., etc). Rule 3.2. Each of s, gs, f, gf excludes all heirs except any x ∈ A , though gf excluding fb, fs, cb, cs is not agreed upon. Rule 3.3. Each x ∈ A (except m, [ gm ( s )] , p ) excludes ubs . Rule 3.4. Each x ∈ H excludes all below him in that order, except that fs, cs do so only by taasiib with d, gd . Rule 3.5. Two (not one) d ( s ) , fs ( s ) exclude all gd ( s ) , cs ( s ) if no gs ( s ) , cb ( s ) respectively. Rule 3.6. Each higher residuary in T takes the residue before the lower while each x ∈ T 2 takes the residue together with her male counterpart and each x ∈ T 3 takes the residue in place of her absent male counterpart if d or gd is present.
4 ABUU ABDILQUDDUUS AS-SALAFEE 4. Inheritance exclusion and residuary chart The annotative chart is informative: (i) terminal arrows indicate continuity in the absence of heirs at one level where applicable; (ii) blue colour indicates the set T 1 ; (iii) green, T 2 ; (iv) orange, T 3 ; (v) gold, T 4 ; (vi) pure As - haabul furuudh ( T c ) in ash colour; (vii) members of the ordered set H arranged in hierarchical steps indicating the order; (viii) an indication of the d/gd and fs/cs exclusion relations; and (ix) an indication of the 2 : 1 with respect to s : d , gs : gd , fb : fs and cb : cs . 5. Inheritance of Classes of Heirs Now, we present easy-to-remember ( in shaa - a Allah ) short representation equations for the shares of heirs according to the shariah. 5.1. Father The share of the father is given by: v ( f ) = 1 6 + R,
INHERITANCE: MATHEMATICAL PRESENTATION 5 where R = 0 ( fard only) if at least a male descendant (that is any of s , gs ) is present. Otherwise, R is the residue (if any) whether f is the only heir or with mother ( ta ′ asiib only); or inherits with female descendants (that is any of d , gd ; fard and ta ′ asiib ). 5.2. Mother Let A be the set whose elements consist of any of the descendants ( s , d , gs , gd ) or at least two from brothers and/or sisters at the same level ( fb ( s ), fs ( s ), cb ( s ), cs ( s ), ub ( s ), us ( s ), fb + fs , cb + cs , ub + us ). Then � 1 i f A = φ, 3 v ( m ) = 1 i f A � = φ. 6 5.3. Spouses Let A be the set whose elements consist of any of s, d, gs, gd (descendants). Then: � 1 i f A = φ, 4 v ( w ( s )) = 1 i f A � = φ. 8 Given same conditions respectively, the h inherits 2 times v ( w ( s )), that is 1 2 , 1 4 respectively. 5.4. Children The shares of children are as follows: � R [2 : 1] i f x = [ s ( s ) : d ( s )] , s ( s ) � = 0 , v ( x ) = � 1 2 , 2 � 3 ; R 1 [2 : 1] i f x = ( d, d ( s ); [ gs ( s ) : gd ( s )]) , with R 1 being residue of 2 3 if gs ( s ) = 0 (that is 1 6 or 0), otherwise R 1 = R . However, if gs ( s ) = 0 (at same level as gd ( s )) and x = d ( s ), then gd ( s ) seek at-ta’asiib with lower-level gs ( s ) to inherit. The gs ( s ) , gd ( s ) at the next level of descendancy inherit as the s ( s ) , d ( s ) in the previous. 5.5. Siblings The shares of siblings are as follows: R [2 : 1] i f x = [ fb ( s ) : fs ( s )] , fb ( s ) � = 0 , � 1 2 , 2 � v ( x ) = 3 ; R 1 [2 : 1] i f x = ( fs, fs ( s ); [ cb ( s ) : cs ( s )]) , � 1 6 , 1 � i f x = ( ubs, ubs ( s )) , 3 with R 1 being residue of 2 3 if cb ( s ) = 0, otherwise R 1 = R . The cb ( s ) , cs ( s ) at the next proximity level inherit as the fb ( s ) , fs ( s ) in the previous. 5.6. Grandfather The gf inherits as follows:
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