Q.Write the Boolean expression for the borrow of the half subractor.

A half subtractor is the combinational circuit that performs the subtraction of one binary bit from another. It has two inputs — the minuend A and the subtrahend B — and two outputs, the difference and the borrow.

The difference is 1 whenever the two input bits are different, exactly like the sum of a half adder, so $\text{Difference} = A \oplus B$. The borrow is generated only when a larger bit is subtracted from a smaller one, that is when $A = 0$ and $B = 1$; this gives $\text{Borrow} = \overline{A} \cdot B$. The word half is used because the circuit does not take into account any borrow produced by a previous, lower-order stage — that extension is the job of the full subtractor. …