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The most basic digital arithmetic circuit is the addition of two binary digits. A combinational circuit that performs the arithmetic addition of two bits is called a half-adder. One that performs the addition of three bits (two significant bits and a previous carry) is called a full-adder. The name of the former stems from the fact that two half-adders are needed to implement a full-adder.

The input variables of a half-adder are called the augend and addend bits. The output variables the sum and carry. It is necessary to specify two output variables because the sum of 1 + 1 is binary 10, which has two digits. We assign symbols x and y to the two input variables, and S (for sum) and C (for carry) to the two output variables. The truth table for the half-adder is shown in Fig. 1-16(a). The C output is 0 unless both inputs are 1. The S output represents the least significant bit of the sum. The Boolean functions for the two outputs can be obtained directly from the truth table:

S=x'y+xy'=x?y

 C = xy

The logic diagram is shown in Fig. 1-16(b). It consists of an exclusive-OR gate and an AND gate.

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