4 – It’s only logical

• Describe three logical operators, and how they are used to form logical expressions
• Use logic gates to construct logic circuits, and associate these with logical operators and expressions
• Describe how hardware is built out of increasingly complex logic circuits

• Logical operators (NOT, AND, OR),
• logical expressions
• truth values (true, false)

• truth tables
• logic gates

• logic circuits
• hardware components

Download the worksheet below to investigate the logic behind a couple of gear mechanisms.

Download Starter Activity WORKSHEET

In 1854, George Boole published The Laws of Thought.

The book didn’t really capture how we think.

It was an effort to represent logic and reasoning as mathematical operations.

Arithmetic operations, such as addition, operate on numbers.

The result, the value of an arithmetic expression, is also a number.

Logical operations operate on statements that are true or false.

The result, the value of a logical expression, is either true or false.

There are three fundamental logical operations:

not  (inversion)

and  (conjunction)

or  (disjunction)

NOT

raining	not raining
false	true
true	false

The statement not A is true when A is false and vice versa.

AND

motion	activated	motion and activated
false	false	false
false	true	false
true	false	false
true	true	true

The statement A and B is true when both A and B are true.

OR

left	right	left or right
false	false	false
false	true	true
true	false	true
true	true	true

The statement A or B is true when at least one of A or B is true.

Use the worksheet below to practise applying Boolean logic.

Download Activity 1 WORKSHEET

In 1940, Claude Shannon proved that logical expressions and electrical circuits with switches are equivalent.

Logical expressions — logic circuits can be represented using diagrams.

Logical operations — logic gates can be represented using symbols.

We use this abstract symbols because we are not interested in the details of the circuit.

Use the worksheet below to create and investigate a few logic circuits.

Download Activity 2 WORKSHEET

[Hardware components] are built of Boolean logic blocks, and the latter are based on simple logical functions, such as And, Or, and Invert. These logical functions are implemented by switches […] and these switches control a physical substance, such as water or electricity, which is used to send one of two possible signals from one switch to another: 1 or 0.

The Pattern on the Stone
by Daniel Hillis (1998)

Logic gates can be combined to form logic circuits that perform a specific function.

Logic circuits can be combined to form increasingly complex hardware components.

Logic describes the function of hardware components.

Logic is binary.

Propositions are either true or false.

This is why, in hardware, instructions and data alike are represented as binary digits.

A circuit that determines the winner in a rock-paper-scissors game

Circuits that add single binary digits

This Google Doodle was created to celebrate the bicentennial of Boole’s birth.

These are the letters that ‘light up’ (are true) when x is true and y is false.

Which letters will light up when x is false and y is true?

Which letters will light up when both x and y are false?

Which letters will light up when both x and y are true?

In this lesson, you…

• Described the NOT, AND, and OR logical operators, and how they are used to form logical expressions
• Used logic gates to construct logic circuits, and associated these with logical operators and expressions
• Described how hardware is built out of increasingly complex logic circuits

Next lesson, you will…

• Define artificial intelligence and machine learning
• Explore examples of where they are being applied
• Teach a machine how to recognise different types of images
• Discuss moral issues associated with these technologies

• Complete the Starter Activity – Gears, again! and upload the Word document to www.bournetolearn.com.

• Complete Activity 1 – Boolean logic and upload the Word document to www.bournetolearn.com.

• Complete Activity 2 – Logic Circuits and upload the Word document to www.bournetolearn.com.