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THE INHERITANCE OF LEGAL HEIRS: MATHEMATICAL PRESENTATION

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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), s1, s2, s3, s4 – 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, s1, s2, fu, hu, s3, s4, q, m, gm(s), p, ubs}. Noting that gm(s) counts 3 while q, p and ubs each counts 2, the cardinality

• f 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 T1 = {s, gs, f, gf, fb, cb, s1, s2, fu, hu, s3, s4} This set consists of residuaries (all heirs who inherit residue) by self. 2.4. al-′Asabatu bil ghayr T2 = {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 T3 = {fs, cs} This is the set of residuaries with others. In the absence of their male counterparts, they take their places in T1 as residuaries, if d or gd is present. 2.6. al-′Asabatu bis sabab T4 = {q} This is the set of residuaries by circumstance.

INHERITANCE: MATHEMATICAL PRESENTATION 3

2.7. All ′Asabah T = ∪Ti = {s, [d], gs, [gd], f, gf, fb, [fs], cb, [cs], s1, s2, fu, hu, s3, s4, 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 T1 − G = {s, gs, fb, cb, s1, s2, fu, hu, s3, s4} 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], s1, s2, fu, hu, s3, s4} 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
• nly 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 ∈ T2 takes the residue together with her male counterpart and each x ∈ T3 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 T1; (iii) green, T2; (iv) orange, T3; (v) gold, T4; (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

• r 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 v(m) =

• 1

3

if A = φ,

1 6

if A = φ. 5.3. Spouses Let A be the set whose elements consist of any of s, d, gs, gd (descendants). Then: v(w(s)) =

• 1

4

if A = φ,

1 8

if A = φ. 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: v(x) =

• R[2 : 1]

if x = [s(s) : d(s)], s(s) = 0, 1

2, 2 3; R1[2 : 1]

• if x = (d, d(s); [gs(s) : gd(s)]),

with R1 being residue of 2

3 if gs(s) = 0 (that is 1 6 or 0), otherwise R1 = 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: v(x) =      R[2 : 1] if x = [fb(s) : fs(s)], fb(s) = 0, 1

2, 2 3; R1[2 : 1]

• if x = (fs, fs(s); [cb(s) : cs(s)]),

1

6, 1 3

• if x = (ubs, ubs(s)),

with R1 being residue of 2

3 if cb(s) = 0, otherwise R1 = 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:

6 ABUU ABDILQUDDUUS AS-SALAFEE

(i) gf inherits like f as in (1). This is the most correct opinion, accord- ing to scholars. (ii) Suppose gf inherits with any of fb, fs, cb, cs (this is not agreed upon!), then gf inherits as follows: let A be the set whose elements consist of any of d, gd, m, gm(s), p that are present and not excluded. Also let y be the share of fb or cb. Then: v(gf) = 1

3, y

• if A = φ,

1

6, 1 3R, y

• if A = φ.

When A = φ, gf has two options ( 1

3 of the whole estate or y), other-

wise he has three options ( 1

6 of the whole estate, 1 3 of the residue or

y); where y means that he reckons as a fb or a cb as applicable (this is called muqasamah). In all cases, he is given the best option. 5.7. Grandmothers All same-level gm(s) take and share 1/6 of the estate if not excluded. They are excluded by m and nearer-level gm(s). 5.8. The Set B = {s1, s2, fu, hu, s3, s4} Each member of the set inherits the residue, in that order, if not excluded (See Rule 3.4). 5.9. The Duo, q Each of duo mu′tiq or mu′tiqah, q (and their asabatu bin nafs), inherits the residue if not excluded.

• 6. Some Special Cases

These are: 6.1. al-′Umariyyataan: {f, m, p} In this case p takes his/her fard, while m takes 1

3 of the residue after the

fard of p, then f takes the remaining. This case does not apply to gf. He takes his fard ( 1

6) only, since m is closer to the deceased.

6.2. al-Himaariyyah: {h, m[gm(s)], ubs(s), fb(s)[fb(s) + fs(s)]} This is also known as al-Musharakah. Here the fixed shares exhaust the estate leaving the ′asabah with nothing! This is the correct opinion due to the statement of the prophet (sallaLahu ′alayhi wasallama), though some opine that only h and m[gm(s)] take their furuudh which are 1

2, 1 6

respectively., while others share the residue ( 1

3) equally.

INHERITANCE: MATHEMATICAL PRESENTATION 7

6.3. al-Akdariyyah: {h, m, gf, fs[cs]} The fixed shares are 1

2, 1 3, 1 6 and 1 2 respectively. leading to al −′ awl given

by 3+2+1+3

6

= 9

• 6. Hence rebasing, we have 3

9, 2 9, 1 9 and 3 9 respectively. Then

the shares of gf and fs(cs) are pulled together for al-muqassamah, that gf and fs(cs) share 4

9 in ratio 2:1 (See gf).

6.4. al-Mu′aadah: {gf, fb(s)[fs(s)], cb(s)[cs(s)]} The cb(s)[cs(s)] are reckoned as fb(s)[fs(s)] respectively. Then gf takes his ‘best’ share (by muqassamah) and fb(s)[fs(s)] and cb(s)[cs(s)] inherit as if the gf was never present (See No. 5.5 - Siblings). Note that al-Mu′aadah is necessary only when al-muqassamah is best for the gf. 6.5. Akhun Mubaarak: {d(s), gd(s), gs} The brother gs is referred to as akhun mubaarak due to his bringing benefit to his sisters gd(s). 6.6. Akhun Mash-uum: {h, fs, cb, cs} The brother cb is referred to as akhun mash-uum due to his denying benefit to his sisters cs who would benefited by al-′awl without him, who has made her an ′asabah.

• 7. Computation

Note that in the computation, one of three situation may arise: 7.1. ar-Radd This is when estate is not exhausted by the heirs. Scholars also differ on whether the left-over will be redistributed among the heirs or given out to non-heirs or baytul-maal. ar-Radd occurs only where there is no ′asabah present. 7.2. al-′Adl This when the estate is just exhausted by the heirs. 7.3. al-′Awl This is when the estate cannot go round the furuudh. In this case, rebas- ing will be required (See 5.3). Here, the presence of an ′asabah makes no difference. Exercises: Indicate the shares of each heir in the sets consisting of: (1) s, gs; (2) gs, gd, ggs; (3) f, gs; (4) f, 2gd, ggd, ggs; (5) f, gf, fb, cb, ub; (6) cb, 5s1; (7) gf, s1; (8) s1, s2, m; (9) s1, d1, s2, d2, s3, d3; (10) gg...gs, fu; (11) s3, fu; (12) w, q; (13) gd, cs, (14) gf, cs Workshop Exercises: Everyone in class gives a practical example.

8 ABUU ABDILQUDDUUS AS-SALAFEE

• Acknowledgment. My sincere appreciation goes to all my shaikhs, those, who

at one time or the other, I have sipped from their fountain of knowledge. Though too numerous to list, I must not forget to mention as-Shaikh Abdul- wahab Muhammad Jaami’ Elesin from whom I studied this subject. May Allah, the Most High, appreciate them all, establish their feet on the truth, and grant them all good here and hereafter. JazaahumuLlahu khayran. Fi- nally, I beg my Lord, the Most High, to accept this little effort as one purely to seek his Exalted Face, and save its reward for me till that day when riches and children shall not profit any soul, make it beneficial, Innahu Semii’un

• Mujeeb. SubhaanaKa Allahumma wabihamdiKa ash-adu an laa ilaaha illa

Anta, astaghfiruKa, wa-atuubu ilayK. References

[1] Tafseerul Qur’aanil ’Adheem by al-Imam al-Haafidh ’Imaadudeen Ismaa’eel Bn ’Umar Bn Katheer (rahimahuLlahu). Published by Resalah Publishers, Damascus, Syria, 1433 (2012). [2] Tayseerul Kareemir Rahman fii tafseer kalaamil Mannaan by al-’Allaamah as- Shaikh AbdurRahman Bn Naasir as-Sa’adee (rahimahuLlahu). Published by Daarul Haythamee, Cairo, Egypt, 1428 (2007). [3] at-Tafseerul muyassar by Group of Scholars. Published by the King Fahd Complex for Printing of the Glorious Quran. [4] at-Tahdheeb fil faraaidh by al-’Allaamah as-Shaikh Mahfuuz Bn Ahmad Bnil Has- san Bn Ahmad al-Kalwadhaanee (rahimahuLlahu). Published by the Daarul Kutubil ’Ilmiyyah, Beirut, Lebanon, 1419 (1998). [5] Sharh ’umadatul ’ahkaa, Combined Volume of al-Imam Ibn Daqeeqil ’Eed, al- ’Allaamah Muhammad Bn Saalih al-’Uthaymeen and al-’Allaamah Abdullah Bn Saalih Aal Basaam (rahimahuLlahu). Published by Daarul Hawzee, Cairo, Egypt, 1421 (2010). [6] Saheeh fiqhis sunnah by as-Shaikh Abuu Maalik Kamaal Bn as-Sayyid Saalim (rahimahuLlahu). Published by Daarut Tawfeeqiyyah li Turaath, Cairo, 1421 (2010). New York Inc. 1983. [7] Manhaajul muslim by as-Shaikh Abuu Bakr Jaabir al-Jazaairee (rahimahuLlahu). Published by Daarul Jawzee, Cairo, 1421 (2010). New York Inc. 1983. [8] Matnur Rahabiyyah wal fawaaiddul jaliyyah fiil Mabaahithil fardhiyyah by al- ’Allaamah as-Shaikh AbdulAzeez Bn Baaz (rahimahuLlahu). Riyadh 1426 (2005). [9] Tashiilul Faraaidh by al-’Allaamah as-Shaikh Muhammad bn Saalih al-’Uthaymeen (rahimahuLlahu). Cairo 1436 (2015). [10] Inheritance in Islam by as-Shaikh Muhammad Imran Muhammad (Available online).

Department of Mathematics University of Ilorin Ilorin, Nigeria E-mail: babalola.ko@unilorin.edu.ng, kobabalola@gmail.com

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