How are arithmetic and memory built from logic gates using adders and flip-flops?
Logic circuits built from gates: the half adder and full adder for binary addition, and the D-type flip-flop as a single-bit memory element, including their truth tables and Boolean expressions.
An OCR H446 answer on logic circuits: the half adder and full adder that perform binary addition (with their sum and carry expressions and truth tables), and the D-type flip-flop that stores a single bit, including how these combine to build arithmetic and memory.
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What this dot point is asking
OCR wants the half adder and full adder (their truth tables and the Boolean expressions for sum and carry, and why the full adder is needed for multi-bit addition), and the D-type flip-flop as a clocked single-bit store. Expect "give the sum and carry expressions" and "explain how a flip-flop stores a bit" questions.
The answer
The half adder
The full adder
The D-type flip-flop
Examples in context
The arithmetic logic unit adds with a chain of full adders, and subtracts by adding the two's complement, so one circuit does both. Registers such as the accumulator and program counter are built from D-type flip-flops, which is why register access is so fast. Counters made from flip-flops drive timing and the program counter's increment. OCR ties this to number representation (the binary the adder processes), to Boolean simplification (designing the gate-level circuits), and to processor architecture (registers and the ALU).
Try this
Q1. Give the Boolean expressions for the sum and carry outputs of a half adder. [2 marks]
- Cue. Sum ; carry .
Q2. State why a full adder, rather than a half adder, is used for all but the least significant bit when adding multi-bit numbers. [2 marks]
- Cue. A full adder has a carry-in, so the carry from the previous column can be included; a half adder has no carry-in and cannot be chained.
Q3. State what happens to a D-type flip-flop's output on the rising edge of the clock. [1 mark]
- Cue. It captures the value on the D input and holds it at Q until the next clock edge.
Exam-style practice questions
Practice questions written in the style of OCR exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
OCR 20206 marksDescribe the function of a half adder, give the Boolean expressions for its sum and carry outputs, and explain why a full adder is needed to add multi-bit numbers.Show worked answer →
Half adder (up to 3): a half adder adds two single binary digits and , producing a sum and a carry. Its truth table: gives sum 0 carry 0; and give sum 1 carry 0; gives sum 0 carry 1.
Expressions (up to 2): the sum is (XOR) and the carry is (AND).
Why a full adder (up to 1): a half adder has no input for a carry coming in from a less significant column, so it cannot be chained to add multi-bit numbers. A full adder adds three inputs (, and a carry-in), so full adders can be cascaded, the carry-out of one feeding the carry-in of the next. Markers reward the half adder's two outputs, the XOR/AND expressions, and the carry-in limitation.
OCR 20225 marksExplain the purpose of a D-type flip-flop, how it stores a bit, and one use of flip-flops in a computer.Show worked answer →
Purpose and operation (up to 3): a D-type flip-flop is a circuit that stores a single bit. It has a data input D and a clock input. On the active (rising) edge of the clock, the value on D is captured and appears at the output Q, and it is then held stable until the next clock edge, regardless of further changes on D between edges. This makes it edge-triggered and synchronous.
Use (up to 2): flip-flops are the building block of registers and other volatile memory, storing the bits of data the processor works on; grouped together they form registers, counters and the basis of static RAM. Markers reward "stores one bit", capture on the clock edge and hold, and a valid use such as registers or counters. A common error is omitting the role of the clock.
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