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Question

Prove that:

(i) a+b+ca1b1+b1c1+c1a1=abc

(ii) (a1+b1)1=aba+b

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Solution

(i) a+b+ca1b1+b1c1+c1a1=abc

LHS=a+b+ca1b1+b1c1+c1a1 {am=1am}=a+b+c1ab+1bc+1ca=a+b+cc+a+babc=(a+b+c)abc(a+b+c)=abc=RHS

(ii) (a1+b1)1=aba+b

LHS=(a1+b1)1=(1a+1b)1=(b+aab)1 {am=1am}=aba+b=RHS


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