Spark
Loading...

Boolean Algebra, Logic Gates, Truth Tables, De Morgan's Law

← Builds on Diodes and LEDs (PN Junction Basics)

Terms you will need

  • boolean algebraBoolean algebra is a type of math that uses only two values, true (1) and false (0), and works with operations like AND, OR, and NOT to describe logical relationships.
  • logic gatesLogic gates are electronic devices that perform basic logical operations, like AND, OR, and NOT, on binary inputs to produce a single binary output.
  • truth tablesA truth table is a chart that shows all possible combinations of input values and the corresponding output for a logical operation or function.
  • and gateAn AND gate is a logic gate that outputs true (1) only when all its inputs are true (1).
  • or gateAn OR gate is a logic gate that outputs true (1) if at least one of its inputs is true (1).
  • not gateA NOT gate is a logic gate that outputs the opposite of its input. If the input is true (1), the output is false (0), and vice versa.

How this topic runs

  1. 1Start here
  2. 2How it works
  3. 3At competition depth

Boolean algebra is a branch of mathematics that deals with logical operations and binary variables, which can only have two values: true (1) or false (0). It is fundamental to the design and analysis of digital circuits and computer systems. Unlike elementary algebra, which uses numbers and arithmetic operations, Boolean algebra uses logical operators such as AND, OR, and NOT to manipulate truth values. This algebra was introduced by George Boole in the 19th century and later became essential in the development of digital electronics and computer science. Logic gates, which are physical devices that implement Boolean operations, form the building blocks of digital circuits. Truth tables are used to represent the outputs of these operations for all possible combinations of inputs. De Morgan's Law, a key principle in Boolean algebra, provides a way to simplify and transform logical expressions.

Figures (10)

A logic circuit diagram for a 4-bit carry lookahead binary adder design using only the AND, OR, and XOR logic gates
A synchronous 4-bit up/down decade counter symbol (74LS192) in accordance with ANSI/IEEE Std 91-1984 and IEC Publication 60617-12 [missing "C3" at pin 11]
Circuit diagram: NOR(NOR(NOR(A, A), NOR(B, B)), NOR(A, B))
Circuit diagram: NAND(NAND(NAND(A, A), NAND(B, B)), NAND(A, B))
CMOS diagram of a NOT gate, also known as an inverter. MOSFETs are the most common way to make logic gates.
4-to-1 multiplexer
Figure 2. Venn diagrams for conjunction, disjunction, and complement
Animation of how an SR NOR gate latch works
The 7400 chip, containing four NANDs. The two additional pins supply power (+5 V) and connect the ground.
Figure 21.1 Electric circuits in computers and servers allow large amounts of data to be quickly and accurately analyzed. (credit: WOCinTech Chat/Flickr)

Boolean algebra works with true (1) and false (0) values, using operations like AND, OR, and NOT to combine or change these values. Logic gates are physical devices that perform these operations, like switches that follow strict rules to produce outputs based on inputs. Truth tables list all possible input combinations and their results, helping to understand how logic gates behave.

A student should picture it as a way to simplify and analyze decisions made by circuits using only two states, true or false, and how those decisions combine to create more complex behaviors.

Key Points

  • Boolean algebra is a branch of algebra that deals with the truth values true and false, usually denoted by 1 and 0, and uses logical operators such as conjunction (and), disjunction (or), and negation (not).
  • Logic gates are devices that perform a Boolean function, a logical operation performed on one or more binary inputs that produces a single binary output.
  • A truth table is a mathematical table used in logic that sets out the functional values of logical expressions on each of their functional arguments, showing the result of the logical operation for each combination of values taken by their logical variables.
  • An AND gate is a logic gate that produces a value of true if both of its operands are true, and a value of false if at least one of its operands is false.
  • An OR gate is a logic gate that produces a value of true if at least one of its operands is true, and a value of false if both of its operands are false.
  • A NOT gate is a logic gate that produces a value of true if its operand is false and a value of false if its operand is true.

Terms

Tap a term for a plain-language explanation.

Sources & licensing(4)