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

step 1 - find the intermediate values P and Q

step 2 - find intermediate values R use P and Q

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

final result

Type 2 - write logic expressions from given logic circuits

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

• 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

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)

Type 4 - Produce a logic circuit and truth table from real work case

A safety system uses three inputs to a logic circuit. An alarm, X, sounds if input A represents ON and input B represents OFF; or if input B represents ON and input C represents OFF.

Type 5 Produce the working space (optional)

Consider the logic statement:

((A NOR B) AND C) NAND (A OR NOT B)

1. Draw a logic circuit to represent the given logic statement.
2. Complete the truth table for the given logic statement.

first - break down the logic statement

1. P = (A NOR B)
2. Q = (A OR NOT B)
3. R = (P AND C)