ALevel-CS Chapter 04

Logic gates and logic circuits

4.01 Boolean logic and problem statements

Key Terms

  • Logic proposition - a statement that is either TRUE or FALSE
  • Problem statement - an informal definition of an outcome which is dependent on one logic proposition or a combination of two or more logic propositions

Boolean logic and problem statements

Logic proposition can have only one of the two alternative Boolean logic values:: TRUE or FALSE.

4.02 Boolean operators

Key Terms

  • Logic expression - logic propositions combined using Boolean operators, which may be equated to a defined outcome

Boolean operators

The three basic Boolean operators are AND, OR and NOT.

  • A AND B is TRUE if A is TRUE and B is TRUE
  • A OR B is TRUE if A is TRUE or B is TRUE
  • NOT A is TRUE if A is FALSE.
  • A NAND B is TRUE if A is FALSE or B is FALSE
  • A NOR B is TRUE if A is FALSE and B is FALSE
  • A XOR B is TRUE if A is TRUE or B is true but not both of them.

Truth tables

The truth table for the AND operator

4.04 Logic circuits and logic gates

Key Terms

  • Logic gate - a component of a logic circuit that has an operation matching that of a Boolean operator

Logic circuits and logic gates

Example Constructing a logic circuit from a problem statement or logic expression

Consider the following problem statement:: A bank offers a special lending rate to customers subject to certain conditions. To qualify, a customer must satisfy certain criteria.

  • The customer has been with the bank for two years.
  • Two of the following conditions must also apply
  • the customer is married
  • the customer is aged 25 years or older
  • the customer’s parents are customers of the bank.

To convert this statement to a logic expression using symbols we can choose:

  • let A represent an account held for two years
  • let B represent that the customer is married
  • let C represent that the customer is aged 25 years or older
  • let D represent that the customer’s parents have an account.

The logic expression can then be written as::

A AND (((B AND C) OR (B AND D)) OR (C AND D))

This could alternatively be presented with an outcome::

Special_rate IF A AND (((B AND C) OR (B AND D)) OR (C AND D))

alternatively as

X = A AND (((B AND C) OR (B AND D)) OR (C AND D))

  • NOT Gate

  • AND Gate

  • OR Gate

  • NAND Gate (NOT AND)

  • NOR Gate (NOT OR)

  • XOR Gate (XOR Gate)

  • . represents the AND operation
  • + represents the OR operation
  • a bar (above the letter or letters, e.g. a) represents the NOT operation.
Example Constructing a truth table from a logic expression or logic circuit

Logic circuits, logic expressions, truth tables and problem statements

Type 1 - produce a truth table for a given logic circuit

题目：找出对应逻辑图(logic circuit)的真值表(truth table)

Step 1 - find the intermediate values P and Q

找出对应逻辑图(logic circuit)的真值表(truth table)

Step 2 - find intermediate values R use P and Q

找出对应逻辑图(logic circuit)的真值表(truth table)

Step 3 - find final part X use intermediate R and C

找出对应逻辑图(logic circuit)的真值表(truth table)

Final result

找出对应逻辑图(logic circuit)的真值表(truth table)

Type 2 - write logic expressions from given logic circuits

Example 1

  • logic gate1:: (A AND B)
  • logic gate2:: (B OR C)
  • final result:: (A AND B) XOR (B OR C)
Example 2

  • logic gate 1:: (A NAND C)
  • logic gate 2:: (B AND C)
  • logic gate 3:: (logic gate 1) NOR A ~ ((A NAND C) NOR A)
  • logic gate 4:: ((A NAND C) NOR A) OR (B AND C)

Type 3 - produce a logic circuit and a truth table from given logic expression

Example 1

Given logic expression: (A XOR C) OR (NOT C NAND B)

first step - A XOR C

second step - NOT C NAND B

third step - combine

fourth step - truth table

Type 4 - produce a logic expression and logic circuit from given truth table (optional)

Example 1

logic circuit

Example 2

logic circuit

Example 3

(NOT A AND NOT B AND NOT C) 
(A AND NOT B AND NOT C) 
(A AND B AND NOT C)

final logic expression

(NOT A AND NOT B AND NOT C) 
OR (A AND NOT B AND NOT C) 
OR (A AND B AND NOT C)