Boolean algebra show that (a+b)(b+c)(c+a)=ab+bc+ca
Answers
Question:If a, b,c are positive then show that bc/b+c+ca/c+a+ab/a+b<(a+b+c)1/2
Step-by-step explanation:
Answer:
It is proven that (a + b) (b + c) (c + a) = ab + bc +ca
Step-by-step explanation:
Given problem
show that (a+b) (b+c) (c+a) = ab+bc+ca by using Laws of Boolean Algebra
Take LHS = (a+ b) (b+c) (c+a )
[ from commutative law A +B = B+ C ⇒ (b+ c) = (c + b) ]
= (a + b) (c + b) (c + a)
[ from distributive law A+ (B.C ) = (A+B) (A+C) ⇒ (b+c) (a+c) = (c +ab) ]
= (a+b) ( c+ ab)
= a(c+ab) + b(c+ ab)
= ac + aab + bc + abb
[ from identity law A.A = A ⇒ a.a = a and b.b = b ]
= ac + ab + bc +ab
= ac + bc + ab + ab
[ from identity law A.A = A ⇒ ab . ab = ab ]
= ac + bc + ab = ab + bc +ca = RHS
LHS = RHS
therefore it is proven that (a + b) (b + c) (c + a) = ab + bc +ca