Consider the Boolean operator $$ \ne $$ with the following properties:
$$x \ne 0 = x,\,\,x \ne 1 = \overline x ,\,\,x \ne x = 0$$ and $$x \ne \overline x = 1.$$ Then $$x \ne y$$ is equivalent to

Let, $${x_1} \oplus {x_2} \oplus {x_3} \oplus {x_4} = 0$$ where $${x_1},\,{x_2},\,{x_3},\,{x_4}$$ are Boolean Variables, and $$ \oplus $$ is the $$XOR$$ operator.

Which one of the following must always be TRUE?

Let $$ \ne $$ be a binary operator defined as $$X \ne Y = X' + Y'$$ where $$𝑋$$ and $$𝑌$$ are Boolean variables. Consider the following two statements. $$$\eqalign{ & \left( {S1} \right)\,\,\,\,\,\,\,\left( {P \ne Q} \right) \ne R = P \ne \left( {Q \ne R} \right) \cr & \left( {S2} \right)\,\,\,\,\,\,\,Q \ne R = R \ne Q \cr} $$$

Which of the following is/are true for the Boolean variables $$𝑃, 𝑄$$ and $$𝑅$$?

Consider the following combinational function block involving four Boolean variables $$x, y, a,$$
$$b$$ where $$x, a, b$$ are inputs and $$y$$ is the output.
Which one of the following digital logic blocks is the most suitable for implementing this function?