By including which logical gate, binary adder circuit is used for both binary…

An adder-subtractor circuit reuses a single binary adder for both operations by placing a controlled inverter between the second operand and the adder's inputs. A controlled inverter is built from Ex-OR (XOR) gates because XOR with one input tied to a control line C reproduces B unchanged when C = 0 (B XOR 0 = B) and reproduces the complement of B when C = 1 (B XOR 1 = B').

Applying this to the circuit: each bit of operand B is passed through an Ex-OR gate whose second input is a mode-control line M, and M also feeds the adder's carry-in.

1. M = 0: every Ex-OR gate outputs B unchanged, so the circuit computes A + B (binary addition).

2. M = 1: every Ex-OR gate outputs B' (the one's complement of B) and the carry-in becomes 1, so the circuit computes A + B' + 1, which equals A plus the two's complement of B, i.e. A - B (binary subtraction).

By including the Ex-OR gate ahead of the adder's inputs, one control line M therefore switches the same adder between addition and subtraction.

Checking this against the other gate types confirms none of them provides the required pass-at-0/complement-at-1 behaviour under the standard shared mode-line convention:

• AND gate (Y = A.B): ANDing an operand with a control bit of 0 always forces the output to 0, so it can never reproduce the operand unchanged - it cannot act as a pass-or-complement switch.

• OR gate (Y = A + B): ORing an operand with a control bit of 1 always forces the output to 1, so it too cannot reproduce the operand unchanged for one control value and its complement for the other.

• Ex-AND gate (Y = A.B + A'.B'): this gate also switches between pass-through and complement across its two control states, but with the opposite polarity to what the circuit needs - it outputs B' when the control is 0 and B when the control is 1 - so under the standard single mode-line M convention (where M = 1 must both complement B and set carry-in to 1 for subtraction) it cannot serve as the required pass-at-0/complement-at-1 switch without redesigning the control-line convention.

Only the Ex-OR gate's truth table (output = B when the control is 0, output = B' when the control is 1) meets the controlled-inversion requirement, which is why including an Ex-OR gate lets a binary adder circuit perform both addition and subtraction.