Half Adders, Full Adders and D-type Flip Flops

Overview

Recall questions on half adders and full adders with their truth tables, expressions and gate arrangements, ripple carry adders and propagation delay, tracing a four-bit addition, and D-type flip flops, edge triggering, toggling and their uses.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   What is a half adder for?

   Model answer

   It adds two single binary digits together. It has two inputs, the bits A and B being added, and two outputs: the sum bit S and the carry bit C. It is the basic building block for binary addition in the arithmetic logic unit.

2. 2

   Give the truth table for a half adder.

   Model answer

   Inputs A and B, outputs sum S and carry C. A=0, B=0 gives S=0, C=0. A=0, B=1 gives S=1, C=0. A=1, B=0 gives S=1, C=0. A=1, B=1 gives S=0, C=1, because 1 + 1 in binary is 10, so the sum digit is 0 and 1 is carried.

3. 3

   Describe the gate arrangement of a half adder.

   Model answer

   The two inputs A and B each split at a junction and feed two gates in parallel. Both go into an XOR gate whose output is the sum S, and both also go into an AND gate whose output is the carry C. That is the whole circuit: two gates, two inputs, two outputs. XOR gives the sum because the sum bit is 1 exactly when the inputs differ, and AND gives the carry because a carry is produced only when both inputs are 1.

4. 4

   What is the limitation of a half adder?

   Model answer

   It has no input for a carry coming in from a previous column. It can therefore only ever add the two least significant bits of a pair of numbers; for any higher column there may be a carry from the column to the right, and the half adder has nowhere to accept it. That is why half adders cannot be chained to add multi-bit numbers.

5. 5

   What is a full adder, and what are its inputs and outputs?

   Model answer

   A full adder adds three single bits: the two bits being added, A and B, and a carry-in, Cin, from the previous column. It has two outputs, the sum bit S and the carry-out Cout, which is passed on to the next column. The three inputs are what distinguish it from a half adder.

6. 6

   Give the truth table for a full adder.

   Model answer

   With inputs A, B, Cin and outputs S and Cout: 0,0,0 gives S=0, Cout=0. 0,0,1 gives S=1, Cout=0. 0,1,0 gives S=1, Cout=0. 0,1,1 gives S=0, Cout=1. 1,0,0 gives S=1, Cout=0. 1,0,1 gives S=0, Cout=1. 1,1,0 gives S=0, Cout=1. 1,1,1 gives S=1, Cout=1. Read as arithmetic, the sum bit is 1 when an odd number of the three inputs is 1, and the carry-out is 1 when two or more of them are 1.

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