10.1 Boolean Logic

Standard Logic Gates

1. The 6 Standard Logic Gates

Each gate has a specific logic rule and a corresponding truth table showing every possible input/output combination.

NOT Gate

Output is the inverse of the input.

A	OUT
0	1
1	0

AND Gate

Output is 1 only if BOTH inputs are 1.

A	B	OUT
0	0	0
0	1	0
1	0	0
1	1	1

OR Gate

Output is 1 if AT LEAST ONE input is 1.

A	B	OUT
0	0	0
0	1	1
1	0	1
1	1	1

NAND Gate (NOT AND)

Output is 0 only if BOTH inputs are 1.

A	B	OUT
0	0	1
0	1	1
1	0	1
1	1	0

NOR Gate (NOT OR)

Output is 1 only if BOTH inputs are 0.

A	B	OUT
0	0	1
0	1	0
1	0	0
1	1	0

XOR Gate (Exclusive OR)

Output is 1 if inputs are DIFFERENT.

A	B	OUT
0	0	0
0	1	1
1	0	1
1	1	0

2. Logic Algebra Symbols

In exams, you may see logic represented as equations:

AND: $A \cdot B$ or $A \text{ AND } B$
OR: $A + B$ or $A \text{ OR } B$
NOT: $\bar{A}$ or $\text{NOT } A$

⚠️ Exam Note: When drawing a logic circuit from an expression like $X = (\text{A AND B}) \text{ OR (NOT C)}$, always work from the inside of the brackets outwards.

Logic in Real-Life

1. Sensors as Logic Inputs

In automated systems, sensors provide the binary input (0 or 1) based on a threshold value. For example:

Temperature:
1 = Too Hot
0 = Normal

Pressure:
1 = Pressed
0 = Not Pressed

Light:
1 = Dark
0 = Bright

Switch:
1 = Closed (On)
0 = Open (Off)

2. Worked Scenario: The Safety Alarm

The Problem: A chemical plant needs an alarm (X) to sound if:

The Temperature (T) is too high (T=1) AND the Pressure (P) is too high (P=1).
OR if the Emergency Switch (S) is pressed (S=1).

Step 1: Map the Logic

IF (T=1 AND P=1) OR (S=1) THEN X=1

Step 2: Create the Expression

X = (T AND P) OR S

Step 3: The Truth Table

This table shows when the alarm will actually sound based on the sensors:

T	P	S	X (Alarm)
0	0	0	0
0	0	1	1 (Switch pressed)
1	1	0	1 (Temp & Press high)
1	1	1	1 (All triggered)

3. Common Real-Life Examples

Street Lighting: Light Sensor (Dark) AND Timer (On) ➔ Lamp On.
Bank Vault: Key 1 (Turned) AND Key 2 (Turned) ➔ Door Opens.
Microwave: Timer (Not 0) AND Door (Closed) ➔ Start Cooking.

⚠️ Exam Note: Always read the "Conditions" carefully. Sometimes an exam will say "Alarm sounds if the window is NOT closed." If Closed = 1, you must use a NOT gate on that input.

Logic Expressions

1. Understanding the Syntax

In IGCSE, expressions can be written in two ways. You must be comfortable with both:

Textual: $X = (\text{A AND B}) \text{ OR (NOT C)}$
Symbolic: $X = (A \cdot B) + \bar{C}$

2. Worked Example: Building a Truth Table

Scenario: Create a truth table for the expression:

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

Step-by-Step Breakdown

1 Identify Inputs: There are 3 inputs (A, B, C). This means there are $2^3 = 8$ possible combinations.

2 Create Intermediate Columns: Solve the brackets first. Create a column for (A AND B) and another for (NOT C).

3 Final Output: Use the OR gate to combine the two intermediate columns.

The Resulting Truth Table

A	B	C	A AND B	NOT C	X
0	0	0	0	1	1
0	0	1	0	0	0
0	1	0	0	1	1
0	1	1	0	0	0
1	0	0	0	1	1
1	0	1	0	0	0
1	1	0	1	1	1
1	1	1	1	0	1

3. Common Logic Gates Combinations

NAND Logic: $\text{NOT (A AND B)}$ is the same as saying "Output 0 only when A and B are 1".
NOR Logic: $\text{NOT (A OR B)}$ is the same as saying "Output 1 only when A and B are both 0".

⚠️ Exam Tip: Always use Intermediate Columns. Even if the question doesn't ask for them, drawing them on your scrap paper prevents simple logic errors that ruin the entire final output column.