Number of Elements in A United Set

axiom 1: m(A) = m(A)
m{a,b} = 4, because {a,b} =_num {∅,{a},{b},{a,b}}
axiom 2: m(AUB) = m(A)+m(B)-m(A∩B)
m{{a,b,c}U{b,c,d}} = 12, because {{a,b,c}U{b,c,d}}= _num m{∅,{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c}}U{∅,{b},{c},{d},{b,c},{b,d},{c,d},{b,c,d}} = _num m{∅,{a},{b},{c},{d},{a,b},{a,c},{b,c},{b,d},{c,d},{a,b,c},{b,c,d}} = _num m{(8)+(8)-(4)} = _num m{(12)}.

m(AUBUC) = m(A)+m(B)+m(C)-m(A∩B)-m(A∩C)-m(B∩C)+m(AUBUC)

1. m(AUBUC) = m(AU(BUC))
2. m(AU(BUC)) = (m(A))+(m(B)+m(C)-m(A∩B)-m(A∩C)-m(B∩C)+m(AUBUC))
3. m(AU(BUC)) = m(A)+m(BUC)-m(A∩(BUC))
4. m(BUC) = m(B)+m(C)-m(B∩C)
5. m(A∩(BUC) = m((A∩B)U(A∩C))
6. m((A∩B)U(A∩C)) = m(A∩B)+m(A∩C)-m(A∩B∩C)
7. m(AU(BUC)) = m(A)+m(B)+m(C)-m(B∩C)-m(A∩(BUC))  //m(BUC)
8. m(AU(BUC)) = m(A)+m(B)+m(C)-m(B∩C)-(m(A∩B)+m(A∩C)-m(A∩B∩C)) //m(A∩(BUC))
9. m(AU(BUC)) = m(A)+m(B)+m(C)-m(B∩C)-m(A∩B)-m(A∩C)+m(A∩B∩C) 
10. m(AU(BUC)) = m(A)+m(B)+m(C)-m(A∩B)-m(A∩C)-m(B∩C)+m(A∩B∩C)