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Question

Prove that:

(i) 11+xab+11+xba=1

(ii) 11+xba+xca+11+xab+xcb+11+xbc+xac=1

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Solution

(i) 11+xab+11+xba=1

LHS=11+xab+11+xba=1xbb+xab+1xaa+xba=1xb(xb+xa)+1xa(xa+xb)=xbxa+xb+xaxa+xb=xb+xa(xa+xb)=xa+xb(xa+xb)=1=RHS

(ii) 11+xba+xca+11+xab+xcb+11+xbc+xac=1

LHS=11+xba+xca+11+xab+xcb+11+xbc+xac=1xaa+xba+xca+1xbb+xab+xcb+1xcc+xbc+xac {Put 1=xaa,1=xbb and 1=xcc}=1xa(xa+xb+xc)+1xb(xb+xa+xc)+1xc(xc+xb+xa)=xaxa+xb+xc+xbxa+xb+xc+xcxa+xb+xc=xa+xb+xcxa+xb+xc=1=RHS


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