Boolean Algebra & Logic Circuits (Cambridge (CIE) A Level Computer Science): Flashcards

Three inputs, A, B and carry in (Cin), and two outputs, carry out (Cout) and sum (S).

The XOR of the inputs A, B and Cin.

When at least two of the inputs A, B and Cin are TRUE.

False.

Cout is TRUE when at least two of the three inputs are TRUE.

In a full adder the three inputs are added, so the answer can be 0, 1, 2 or 3.

0 + 1 + 1 = 2, which is 10 in binary, so Cout = 1 and S = 0.

1 + 1 + 1 = 3, which is 11 in binary, so Cout = 1 and S = 1.

Two half adders and an OR gate.

A and B.

Sum1, the output of the first half adder's XOR gate, and Cin.

Carry1 and Carry2 are connected to the inputs of an OR gate, and the output of that gate is the final Cout.

They are digital memory components used in sequential circuits, storing 1 bit of data.

A flip flop stores 1 bit of data.

Counters, control systems and synchronous logic.

A bistable circuit has two stable states. It responds to clock signals and is often edge-triggered.

The rising edge.

Set and Reset.

NOR gates.

Inputs are S (Set), R (Reset) and CLK (Clock) when edge-triggered. Outputs are Q, the current stored value, and NOT(Q), its inverse.

No change. The flip flop holds its current value, which is memory mode.

Reset. Q becomes 0.

Set. Q becomes 1.

It forces Q = 0. Both Q and NOT(Q) are driven to 0.

False.

In NOR-based designs it forces Q = 0. It is in NAND-based designs that this state is undefined.

The flip flop enters memory mode, so Q stays at 1.

It solves the problem of the SR's illegal state by using that condition to toggle, and it uses a clock to control the exact timing of changes.

The JK flip flop uses a clock to control the exact timing of changes.

No change. This is memory or hold mode.

Reset. Q becomes 0.

Set. Q becomes 1.

It toggles the output, so Q becomes NOT(Q).

False.

It toggles the output, so Q becomes NOT(Q).

Simple memory and control circuits.

Toggles, counters and frequency dividers.

Boolean algebra is a mathematical system used to manipulate Boolean values, allowing complex expressions to be made simpler.

It is a more powerful simplification method, and can simplify expressions that Karnaugh maps cannot.

0. An AND only outputs 1 when both inputs are 1, so anything ANDed with 0 is 0 whatever X is.

X. The output is whatever the value of X is.

False.

X AND 1 = X, so the output is whatever the value of X is.

0. X and NOT X always hold opposite values, so one input is always 0, and an AND with a 0 input gives 0.

X. An OR outputs 1 if either input is 1, so ORing with 0 leaves the output as whatever X is.

1. An OR outputs 1 if either input is 1, so a 1 input forces the output to 1 whatever X is.

1. X and NOT X always hold opposite values, so one input is always 1, and an OR with a 1 input gives 1.

A · B, or simply AB.

In Boolean notation the dot for AND is often omitted, so AB means A AND B.

A + B. The plus sign means OR, so A + B reads as A OR B.

As ¬A, as A', or as a line drawn above the variable.

The value is inverted or negated. It is the NOT of that value.

Do A OR B first, then AND the result with C.

A AND (NOT B). B is negated before the AND operation.

The whole expression, not just one part of it.

The outermost line.

NOT (A AND B) is equivalent to (NOT A) OR (NOT B).

NOT (A OR B) is equivalent to (NOT A) AND (NOT B).

It changes: AND becomes OR, and OR becomes AND. Both terms are then negated.

(NOT A) OR (NOT B)

False.

It gives (NOT A) AND (NOT B). The operator changes when the law is applied.

Rewriting logic circuits using only NAND or NOR gates.

De Morgan's Laws allow a circuit to be implemented using only NAND or NOR gates.

They make circuit design more efficient and cost-effective, and they are essential where NAND or NOR logic is cheaper or faster to implement.

In microprocessor and memory device design, for example flash drives.

A Karnaugh map is a tool used for simplifying Boolean algebra expressions, offering a visual method of grouping together expressions with common factors.

The format of the map makes it easy to identify and eliminate redundant terms.

In digital logic design, such as simplifying the logic of digital circuits.

Create the map, group the 1s, simplify each group, then combine the group expressions with OR to get the final simplified expression.

A term in the Boolean expression. Cells are filled with 1s and 0s corresponding to the output of that term.

Split the expression at each OR into subterms, then put a 1 in every cell where that subterm would be TRUE.

Every cell where A is 1, and every cell where B is 1.

A rectangle.

Groups on a Karnaugh map must be rectangular in shape.

The size must be a power of 2, so a group contains 8, 4, 2 or 1 ones.

False.

A group's size must be a power of 2, so it can contain 8, 4, 2 or 1 ones.

As large as possible.

False.

Groups can overlap, so a cell can be part of multiple groups.

The grid wraps round in all directions, so groups can wrap round.

It consists of the variables that remain constant in all the terms in that group.

The simplified expression for a group consists of the variables that remain constant.

It is eliminated from that group's term, because only the variables that stay constant are kept.

A

NOT B

Combine the simplified expressions from each group using OR operations.