Boolean Logic

Boolean Logic

Outline

• Standard logic gate symbols

• Use logic gates to create logic circuits

• Complete a truth table

• Write logic expression

Introduction

• Electronic circuits in computers, solid state drives and controlling devices are made up of thousands of logic gates

• Logic gates take binary inputs and produce a binary output.

• Several logic gates combined together form a logic circuit and these circuits are designed to carry out a specific function.

Logic gate expressions

• Truth table: Shows all the combinations that can be used as input and output.

• Logic expression: Used to represent the logic of the logic gate.

NOT Logic gate

• It is the simplest logic gate with one input and one output.

• It is used to reverse the input to get the output.

• NOT Logic gate

• Truth table

​A	​NOT A
​0	​1
​1	​0

Questions

• You can withdraw money from an ATM only if you enter the correct PIN AND have sufficient funds in your account.

• In a room, the light is designed to turn off when you press the switch, AND it turns on when the switch is released.

• A security light is set to turn on at night when it gets dark AND off during the day when it is light.

• A security light is set to turn on at night when it gets dark and off during the day when it is light.

AND GATE

• Has two input and one output. Output is 1 when both inputs are 1 Otherwise the output is zero.

• AND Gate diagram

• AND gate truth table

• Z = A AND B (Logic notation)

• Z = A.B (Boolean Notation)

​ A	​B	​Z
​0	​0	​0
​0	​1	​0
​1	​0	​0
​1	​1	​1

Multiple Choice

What is the output of an AND gate if both inputs are FALSE?

• TRUE
• UNKNOWN
• MAYBE
• FALSE

Multiple Choice

Given that Input 1 is TRUE and Input 2 is FALSE, what will be the output of the AND gate?

• TRUE
• UNKNOWN
• 1
• FALSE

Multiple Choice

If an AND gate has three inputs (A, B, C) and all inputs are TRUE, what will the output be?

• FALSE
• UNKNOWN
• MAYBE
• TRUE

Multiple Choice

An AND gate will produce a TRUE output only when:

• At least one input is TRUE
• Only one input is TRUE
• All inputs are FALSE
• All inputs are TRUE

OR GATE

• Has two inputs and one output. The output is one when one of the inputs is one or both inputs are one.

• OR Gate diagram

• OR Gate truth table

• Z = A OR B (Logic notation)

• Z = A + B (Boolean notation)

​A	​B	​Z
​0	​0	​0
​0	​1	​1
​1	​0	​1
​1	​1	​1

Multiple Choice

Which of the following inputs will result in a TRUE output from an OR gate?

• FALSE, FALSE; FALSE, FALSE
• Any input where at least one input is TRUE (e.g., TRUE, FALSE; TRUE, TRUE; FALSE, TRUE).
• TRUE, TRUE; TRUE, FALSE
• TRUE, FALSE; FALSE, FALSE

Multiple Choice

What is the output of an OR gate if both inputs are FALSE?

• UNKNOWN
• FALSE
• MAYBE
• TRUE

Multiple Choice

Given that Input 1 is TRUE and Input 2 is FALSE, what will be the output of the OR gate?

• TRUE AND FALSE
• TRUE
• NOT TRUE
• FALSE

Multiple Choice

If an OR gate has three inputs (A, B, C) and at least one input is TRUE, what will the output be?

• TRUE
• UNDETERMINED
• ZERO
• FALSE

NAND Gate (NOT AND)

• Has two inputs and one output. Output is 0 when both inputs are 1. Otherwise the output is 0.

• NAND gate diagram

• NAND gate truth table

• Z = NOT (A AND B).

• Z = A.B

​A	​B	​Z
​0	​0	​1
​0	​1	​1
​1	​0	​1
​1	​1	​0

NOR Logic Gate (NOT OR)

• Has two inputs and one output.

• The only time the output is 1 is when both inputs are 0.

• NOR gate diagram

• Truth table

• Q = NOT(A OR B)

• Q = A+B

​A	​B	Q
​0	​0	​1
​0	​1	​0
​1	​0	​0
​1	​1	​0

XOR gate

• It is also known as EOR gate. It is known as exclusive OR gate. It has two input and one output.

• The output is one when inputs are different and zero when inputs are the same.

• Diagram

​A	​B	​Y
​0	​0	​0
​0	​1	​1
​1	​0	​1
​1	​1	​0

Match

Match the following

AND GATE

OR GATE

NOR GATE

XOR GATE

NOT GATE

• AND GATE

• OR GATE

• NOR GATE

• XOR GATE

• NOT GATE

Multiple Choice

Which of the following is the output of an AND gate when both inputs are 1?

• 2
• 1.5
• 0
• 1

Multiple Choice

Which logic gate produces an output of 1 when at least one of its inputs is 1?

• OR gate
• XOR gate
• NAND gate
• AND gate

Multiple Choice

Which gate inverts the input it receives?

• NOT gate
• AND gate
• OR gate
• NAND gate

Multiple Choice

Which of the following is true for a NOR gate?

• The output is true when both inputs are true.
• The output is false only when both inputs are true.
• The output is true only when both inputs are false.
• The output is true when at least one input is true.

Multiple Choice

If one input of an XOR gate is 0 and the other is 1, what is the output?

• 0
• 1
• 1.5
• 2

Logic Circuit

Combination of several logic gates that are linked together.

Example 01

• Give the logic expression and truth table for the following logic circuit

Answer

• Y = (A OR B ) AND NOT C

​A	​B	​C	NOT C	​A OR B	​(A OR B) AND NOT C
​0	​0	​0	​1	​0	​0
0​	0	1​	0​	0​	0​
0​	1​	0​	1​	1​	1​
0​	1​	1​	0​	1​	0​
1​	0​	0​	1​	1​	1​
1​	0​	1​	0​	1​	0​
1​	1​	0​	1​	1​	1​
1​	1​	1​	0​	1​	0​

Example 02

• Give the logic expression and truth table for the following logic circuit

Answer

• Q =A OR (NOT B AND C)

​A	​B	​C	NOT B	​NOT B AND C	A OR (NOT B AND C)
​0	​0	​0	​1	​0	​0
0​	0	1​	1	1	1​
0​	1​	0​	0	0	0​
0​	1​	1​	0	0	0​
1​	0​	0​	1	0	1
1​	0​	1​	1	1​	1
1​	1​	0​	0	0	1​
1​	1​	1​	0	0	1

Example 03

• Give the logic expression and truth table for the following logic circuit

Answer

• Z = (NOT(P AND Q)) XOR NOT Q) AND R

• A = P AND Q

• B = NOT (P AND Q)

• C = NOT Q

• D = A XOR C

• E = D AND R

​P	​Q	​R	​A	​B	​C	​D	​E
​0	​0	​0	​0	​1	​1	​0	​0
​0	0​	1​	0​	1​	1​	0​	0​
0​	1​	0​	0​	1​	0​	1​	0​
0​	1​	1​	0​	1​	0​	1​	1​
1​	0​	0​	0​	1​	1​	0​	0​
1​	0​	1​	0​	1​	1​	0​	0​
1​	1​	0​	1​	0​	0​	0​	0​
1​	1​	1​	1​	0​	0​	0​	0​

Homework

1. Give the logic expression and truth table for the following logic circuit

2. Give the logic expression and truth table for the following logic circuits

Question

• Complete the truth table for this logic circuit.

​A	​B	​C	​Working Place	​Z
​0	​0	​0	​	​
0​	0​	1​	​	​
0​	1​	0​	​	​
0​	1​	1​	​	​
1​	0​	0​	​	​
1​	0​	1​	​	​
1​	1​	0​	​	​
1​	1​	1​	​	​

Answer

​A	​B	​C	​Working Place	​Z
​0	​0	​0	​	​0
0​	0​	1​	​	1​
0​	1​	0​	​	0​
0​	1​	1​	​	0​
1​	0​	0​	​	​1
1​	0​	1​	​	0​
1​	1​	0​	​	0​
1​	1​	1​	​	0​

Question

• Consider the logic expression:

  Z = (A NAND B) OR NOT (B XOR C)

  Complete the truth table

  from the given logic expression.

​A	​B	​C	​Working Place	​Z
​0	​0	​0	​	​0
0​	0​	1​	​	1​
0​	1​	0​	​	0​
0​	1​	1​	​	0​
1​	0​	0​	​	​1
1​	0​	1​	​	0​
1​	1​	0​	​	0​
1​	1​	1​	​	0​

Answer

​A	​B	​C	​Working Place	​Z
​0	​0	​0	​	​1
0​	0​	1​	​	1​
0​	1​	0​	​	1
0​	1​	1​	​	1
1​	0​	0​	​	​1
1​	0​	1​	​	1
1​	1​	0​	​	0​
1​	1​	1​	​	1​

Question

• Consider this logic circuit.

• 
  
  Write a logic expression for this logic circuit.

  Complete the truth table from the given

  logic circuit.

​A	​B	​C	​Working Place	​Z
​0	​0	​0	​	​0
0​	0​	1​	​	1​
0​	1​	0​	​	0​
0​	1​	1​	​	0​
1​	0​	0​	​	​1
1​	0​	1​	​	0​
1​	1​	0​	​	0​
1​	1​	1​	​	0​