Determine which of the following equations are always

Chapter 3, Problem 3.28

(choose chapter or problem)

Determine which of the following equations are always valid (give an algebraic proof):

(a) \(a^{\prime} b+b^{\prime} c+c^{\prime} a=a b^{\prime}+b c^{\prime}+c a^{\prime}\)

(b) \((a+b)(b+c)(c+a)=\left(a^{\prime}+b^{\prime}\right)\left(b^{\prime}+c^{\prime}\right)\left(c^{\prime}+a^{\prime}\right)\)

(c) \(a b c+a b^{\prime} c^{\prime}+b^{\prime} c d+b c^{\prime} d+a d=a b c+a b^{\prime} c^{\prime}+b^{\prime} c d+b c^{\prime} d\)

(d) \(x y^{\prime}+x^{\prime} z+y z^{\prime}=x^{\prime} y+x z^{\prime}+y^{\prime} z\)

(e) \((x+y)(y+z)(x+z)=\left(x^{\prime}+y^{\prime}\right)\left(y^{\prime}+z^{\prime}\right)\left(x^{\prime}+z^{\prime}\right)\)

(f) \(a b c^{\prime}+a b^{\prime} c+b^{\prime} c^{\prime} d+b c d=a b^{\prime} c+a b c^{\prime}+a d+b c d+b^{\prime} c^{\prime} d\)