Use the properties of addition and multiplication of real numbers given in Prope

Question

Use the properties of addition and multiplication of real numbers given in Properties that (1) Commutativity. a+b=b+a and ab=ba. (2) Associativity. (a+b)+c=a+(b+c) and (ab)c=a(bc). (3) Distributivity. a(b+c)=ab+ac and (a+b)c=ac+bc. (4) Zero. a+0=a=0+a. (5) Unity. a*1=a=1*a (6) Subtraction. The equation a+x=0 has the unique solution x=-a and so a+x=b<=>x=b+(-a)=b-a. (7) Division. If a is not equal to zero, then the equation ax=b has the unique solution x=b/a=b*a^-1. to deduce that, for all real numbers a and b, (i) a x 0 = 0 = 0 x a, (ii) (-a)b= -ab = a(-b) (iii) (-a)(-b) = ab Use the properties of addition and multiplication of real numbers given in Properties that (1) Commutativity. a+b=b+a and ab=ba. (2) Associativity. (a+b)+c=a+(b+c) and (ab)c=a(bc). (3) Distributivity. a(b+c)=ab+ac and (a+b)c=ac+bc. (4) Zero. a+0=a=0+a. (5) Unity. a*1=a=1*a (6) Subtraction. The equation a+x=0 has the unique solution x=-a and so a+x=b<=>x=b+(-a)=b-a. (7) Division. If a is not equal to zero, then the equation ax=b has the unique solution x=b/a=b*a^-1. to deduce that, for all real numbers a and b, (i) a x 0 = 0 = 0 x a, (ii) (-a)b= -ab = a(-b) (iii) (-a)(-b) = ab

Explanation / Answer

(i)ax0=0=0xa
(ii)    (-a)b=[(-1)a]b=(-1)ab=a(-1)b=a[(-1)b]=a(-b) (iii) (-a)(-b)=[(-1)a][(-1)b]=(-1)(-1)ab=(-1)2ab=ab

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