4.3 Boolean Algebra (OCR A Level Computer Science) Flashcards
Exam code: H446
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Define Boolean logic.
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Define Boolean logic.
Boolean logic is the set of principles of logic based on binary values, where values are either TRUE (1) or FALSE (0), and is used to make logical decisions in computer science and electronics.
What is the difference between Boolean logic and Boolean algebra?
Boolean logic refers to the principles of logic based on binary values, while Boolean algebra is a mathematical system used to manipulate Boolean values.
In Boolean logic, the value represents TRUE, and represents FALSE.
In Boolean logic, the value 1 represents TRUE, and 0 represents FALSE.
True or False?
In Boolean expressions, OR operations are evaluated before AND operations unless parentheses are used.
False.
In Boolean expressions, AND operations are evaluated before OR unless parentheses change the order.
Define Exclusive Disjunction (XOR) in Boolean logic.
Exclusive Disjunction (XOR) is a Boolean operator that returns TRUE if and only if the inputs are different, and FALSE if they are the same.
The gate inverts its input, so NOT TRUE = FALSE and NOT FALSE = TRUE.
The NOT gate inverts its input, so NOT TRUE = FALSE and NOT FALSE = TRUE.
Define truth table.
A truth table is a tool used in logic and computer science to visualise the results of Boolean expressions. It lists all possible inputs and the associated outputs for a given Boolean expression.
What does each row in a truth table represent?
Each row in a truth table represents a unique combination of inputs and the associated output for a Boolean expression.
A truth table shows all possible and their associated for a given Boolean expression.
A truth table shows all possible inputs and their associated outputs for a given Boolean expression.
True or False?
Truth tables can be used to check the correctness of logic circuits.
True.
Truth tables allow you to systematically check all possible inputs and outputs of a logic circuit to verify its correctness.
In a truth table for an AND gate, the output is only when both inputs are 1.
In a truth table for an AND gate, the output is 1 only when both inputs are 1.
Define exclusive disjunction (XOR).
An exclusive disjunction (XOR) is a Boolean operation that returns true only when exactly one of the inputs is true. If both inputs are the same, it returns false.
Define Karnaugh map.
A Karnaugh map is a visual tool used to simplify Boolean algebra expressions, making it easier to identify and eliminate redundant terms when designing digital circuits.
What is the main purpose of using a Karnaugh map in digital logic design?
The main purpose of using a Karnaugh map is to simplify Boolean algebra expressions, making it easier to create efficient digital circuits.
In a Karnaugh map, are grouped together in rectangles, and each group must contain a number of cells that is a power of 2.
In a Karnaugh map, 1s are grouped together in rectangles, and each group must contain a number of cells that is a power of 2.
True or False?
Each group in a Karnaugh map can only include cells that are adjacent horizontally or vertically, not diagonally.
True.
Groups in a Karnaugh map must include only cells that are adjacent horizontally or vertically; diagonal grouping is not allowed.
Define Karnaugh Map (KMap).
A Karnaugh Map (KMap) is a diagram used to simplify Boolean expressions with 2 or more inputs by visually grouping terms.
What is the first step in creating a Karnaugh Map for two variables?
The first step is to add each variable, starting with A at the top and B down the side, and to add each possible state for both variables.
To fill a KMap for the expression A V B, you find all cells where either or is 1 and place a 1 in those cells.
To fill a KMap for the expression A V B, you find all cells where either A or B is 1 and place a 1 in those cells.
True or False?
After filling all cells for each variable in a KMap, the map is complete for the given Boolean expression.
True.
Once you have added 1s for all cells where each variable in the Boolean expression is 1, the KMap represents the expression fully.
Define Karnaugh map.
A Karnaugh map is a diagram used to simplify Boolean expressions by organizing truth values into a grid to easily identify groups for minimization.
What is the main goal when grouping 1s in a Karnaugh map?
The main goal is to make the largest possible rectangular groups containing 1s, so the Boolean expression can be simplified as much as possible.
In a Karnaugh map, groups can only be formed in shapes that are and must contain 1s (e.g. 1, 2, 4, or 8).
In a Karnaugh map, groups can only be formed in shapes that are rectangular and must contain powers of two 1s (e.g. 1, 2, 4, or 8).
True or False?
Groups in a Karnaugh map may overlap and can wrap around the edges of the map.
True.
Groups can overlap and wrap around the edges of a Karnaugh map to ensure the largest possible simplification of the Boolean expression.
Define De Morgan's Law.
De Morgan's Law is a strategy for simplifying expressions that include a negation of a conjunction or disjunction. It allows you to invert variables and change AND to OR (or vice versa) when negating logical expressions.
True or False?
NOT(A AND B) is logically equivalent to (NOT A) OR (NOT B).
True.
According to De Morgan's Law, the negation of a conjunction (AND) is equivalent to the disjunction (OR) of the negated terms: NOT(A AND B) = (NOT A) OR (NOT B).
To apply De Morgan's Law, change to OR (or vice versa) and negate each term.
To apply De Morgan's Law, change AND to OR (or vice versa) and negate each term.
Why is De Morgan's Law useful in building microprocessors?
De Morgan's Law is useful because it allows logical expressions to be simplified so that only NAND or NOR gates are needed, which makes microprocessor design easier and more efficient.
Define associative law.
The associative law states that how variables are grouped in an expression with only AND or only OR does not affect the outcome, e.g., (A AND B) AND C = A AND (B AND C) = A AND B AND C.
Define distributive law.
The distributive law explains how AND and OR interact in logic, allowing you to distribute one over the other: A AND (B OR C) = (A AND B) OR (A AND C).
True or False?
A OR (B AND C) is equivalent to (A OR B) AND (A OR C).
True.
According to the distributive law, A OR (B AND C) is the same as (A OR B) AND (A OR C).
The associative law allows us to remove and regroup variables in expressions, so (A OR B) OR C is the same as A OR (B OR C).
The associative law allows us to remove brackets and regroup variables in expressions, so (A OR B) OR C is the same as A OR (B OR C).
Define commutative law.
The commutative law states that the order of the variables does not change the truth value of the expression, such as A AND B being the same as B AND A.
True or False?
A OR B is different from B OR A in Boolean logic.
False.
According to the commutative law, A OR B is the same as B OR A.
The commutative law states that of variables does not change the truth value of the expression.
The commutative law states that the order of variables does not change the truth value of the expression.
Define double negation law.
The double negation law states that applying NOT twice to a variable returns the original variable, so NOT(NOT(A)) = A.
According to the double negation law, NOT(NOT(A)) = .
According to the double negation law, NOT(NOT(A)) = A.
How does the commutative law help when simplifying Boolean expressions, such as in the worked example (A v B) ^ (A v C)?
The commutative law allows you to rearrange variables in expressions (e.g., changing B AND A to A AND B), which helps to group like terms and apply further simplifications when working through Boolean expressions like (A v B) ^ (A v C).
Define D type flip flop.
A D type flip flop is a digital circuit component used to store a single bit of data and is triggered by the rising edge of the clock pulse.
What are the two outputs of a D type flip flop called?
The two outputs are Q and NOT(Q).
A D type flip flop contains stable states, making it a circuit.
A D type flip flop contains two stable states, making it a bistable circuit.
What happens to the output Q of a D type flip flop on the rising edge of the clock pulse if D is high?
If D is high on the rising edge of the clock pulse, Q goes high and NOT(Q) goes low.
True or False?
A D type flip flop can change its output Q at any time when D changes.
False.
A D type flip flop only changes its output Q on the rising edge of the clock pulse, not whenever D changes.
The input of a D type flip flop provides the data to be stored, while the input controls when the state may change.
The D input of a D type flip flop provides the data to be stored, while the CLK input controls when the state may change.
Name one common use of D type flip flops in digital circuits.
D type flip flops are commonly used in shift registers, counters, and memory units.
D type flip flops are often referred to as edge triggered devices.
D type flip flops are often referred to as positive edge triggered edge triggered devices.
Define half adder.
A half adder is a basic digital circuit used to add two single-bit binary numbers, producing a Sum (S) and a Carry out (Cout) output.
What are the inputs and outputs of a half adder circuit?
A half adder circuit has two inputs, labelled A and B, and two outputs: Sum (S) and Carry out (Cout).
In a half adder, the output is produced using an gate, and the output is produced using an XOR gate.
In a half adder, the Carry out (Cout) output is produced using an AND gate, and the Sum (S) output is produced using an XOR gate.
True or False?
The output of a half adder's Sum is created using an AND gate.
False.
The Sum output in a half adder is produced using an XOR gate, not an AND gate.
When both inputs A and B of a half adder are 1, the binary sum is , meaning Carry out is and Sum is
When both inputs A and B of a half adder are 1, the binary sum is 10, meaning Carry out is 1 and Sum is 0
Which logic gates are used to build a half adder circuit?
A half adder circuit uses an XOR gate for the Sum output and an AND gate for the Carry out output.
Define full adder.
A full adder is a logic circuit that adds three binary inputs (A, B, and Cin) and produces two outputs: sum (S) and carry (Cout).
A full adder circuit has inputs (A, B, Cin) and outputs (S, Cout).
A full adder circuit has three inputs (A, B, Cin) and two outputs (S, Cout).
True or False?
The sum output of a full adder is the result of XOR-ing all three inputs: A, B, and Cin.
True.
The sum output (S) of a full adder is calculated by XOR-ing A, B, and Cin. This means S = A ⊕ B ⊕ Cin.
How can you use full adders to add two 4-bit binary numbers?
You can add two 4-bit binary numbers by connecting four full adders in series, with the carry output (Cout) of each adder connected to the carry input (Cin) of the next.
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