Determine whether or not the following expressions are valid, i. e, whether the left and right- hand sides represent the same function: Show the reasoning, using truth table.
a-) x₁'. x₃ + x₁. x₂. x₃ + x₁'. x₂ + x₁. x₂ = x₂x₃ + x₁. x₃ + x₂. x₃ + x₁'. x₂. x₃
b-) x₁. x₃ + x₂. x₃ + x₂x₃ = (x₁ + x₂ + x₃) (x₁ + x₂ + x₃) (x₁' + x₂ + x₃')
c-) (x₁ + x₃) (x₁' + x₂ + x₃) (x₁' + x₂) = (x₁ + x₂) (x₂ + x₃) (x₁' + x₃)