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.

****THIS IS THE QUESTION!!!
Prove the following using above properties, for all real numbers a and b,
(i) a x 0 = 0
(ii) (-a)b= -ab
(iii) (-a)(-b) = ab

Explanation / Answer

1) a*0 = 0 This comes from the fact that 0 + a*0 = a*0 = a(0 + 0) = a*0 + a*0. We then cancel a*0 from each side to get 0 = a*0. 2) (-a)*b = -ab We have ab + -ab = 0 = 0*b = (a-a)b = ab + (-a)*b. Cancel ab from both sides to get (-a)*b = -ab. 3) (-a)(-b) = ab We have (-a)(-b) + -ab = -(a*(-b)) + -ab = -(-(ab)) + -ab = 0, and ab + -ab = 0, so we must have (-a)(-b) = ab.

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