2.1.11 Define the Boolean operators: AND, OR, NOT, NAND, NOR and XOR

DISCLAIMER: the IB requests the logic gates to be represented as circles containing the operator inside, not the complicated sign

true = 1

false = 0

AND

Both inputs are true

Logic Gate Symbols

a AND b are true

Input A

Input B

Output

0

0

0

0

1

0

1

0

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1

1

1

OR

One input at least is true

Input A

Input B

Output

0

0

0

0

1

1

1

0

1

1

1

1

NOT

input is not true

only one input

converse of input is the output

Input A

Output

0

1

1

0

NAND

the inputs are not both true (not AND)

Input A

Input B

Output

0

0

1

0

1

1

1

0

1

1

1

0

NOR

not input A (n)or input B are true

Input A

Input B

Output

0

0

1

0

1

0

1

0

0

1

1

0

XOR

only one input can be true, not both

Input A

Input B

Output

0

0

0

0

1

1

1

0

1

1

1

0

How do logic gates work?

Identify the inputs.

Sketch a truth table.

Enter the inputs and all possible combinations of 0s and 1s. The total number of combinations is 2 to the power of n, where n is the total number of inputs.

In the following columns write the logic operators and how it combines the inputs.

Use the previous columns filled to identify the new combination of 0s and 1s applying the concepts of the logic operators listed above.

Continue until all gates are passed and the output is detected.

How to: Logic Diagram

Identify the inputs. Right them on the left side.

Read the description. Identify the operators and in which way they are combined.

For example (see example 1):

(NOT A) and (A AND B) are the first to draw

These outputs are then combined through (OR)

NOT (A AND B) - this means to negate one of the first outputs

Combine that through the us of OR with the other output to reach the final output.

Step by step combine the operators until a final output is reached.

How to: Logic Tables

Write all possible combinations for the inputs.

use 2 to the power of n, where n is the number of inputs to find how many combinations are possible

Identify the first operation

Use the inputs to complete the first operation according to the instructions given by the operation

e.g. NOT A is the inverse of the input A

Identify the next relevant operation and use the previously constructed columns to fill in the new column.

Repeat until the final output is reached.

e.g. (A AND B) NAND (NOT A)

Consider the inputs A & B

Input A

Input B

0

0

0

1

1

0

1

1

Combine them using AND

A AND B

0 + 0 = 0

0 + 1 = 0

1 + 0 = 0

1 + 1 = 1

Negate A (NOT A)

Input A

NOT A (Inverse)

0

1

0

1

1

0

1

0

Combine (NOT A) NAND (A AND B)

A AND B

NOT A

Output

0

1

1

0

1

1

0

0

1

1

0

1

Logic gates are important in real life, as they can be used in circuits and programs. For example, lets take the example and see where it would be applicable in real life. These could be logic gates regulating whether too much pollution is caused and a warning alarm should be let off.

Example 1: ((NOT A) OR (A AND B)) OR (NOT(A AND B))

Input A

Input B

NOT A

A AND B

D = (NOT A) OR (A AND B)

E = NOT (A AND B)

D OR E = Output C

0

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1

Example 2: ((A OR C) NOR (A AND B)) NAND (B XOR C)

Input A

Input B

Input C

A OR C

A AND B

(A OR C) NOR (A AND B)

B XOR C

NAND

0

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1

Practice Problem - Real Life Example:

If traffic light A is green (on) traffic light, traffic light C can/must be green too. However, traffic light B can not be green then. Thus, if traffic light B is not on, neither is traffic light D.

Hint:

Input A

Input B

Input C

Input D

A AND C

B AND D

(A AND C) NAND (B AND D)

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Created By: Lucie Charlotte Magister Last update: 20/11/2014 Sources:

2.1.11 Define the Boolean operators: AND, OR, NOT, NAND, NOR and XORANDBoth inputs are trueLogic Gate Symbols

OROne input at least is trueNOTinput is not trueNANDthe inputs are not both true (not AND)NORnot input A (n)or input B are trueXORonly one input can be true, not bothHow do logic gates work?2 to the power of n, where n is the total number of inputs.How to: Logic Diagram

How to: Logic Tables

2 to the power of n, where n is the number of inputsto find how many combinations are possiblee.g. (A AND B) NAND (NOT A)Consider the inputs A & BCombine them using ANDNegate A (NOT A)Combine (NOT A) NAND (A AND B)Logic gates are important in real life, as they can be used in circuits and programs. For example, lets take the example and see where it would be applicable in real life. These could be logic gates regulating whether too much pollution is caused and a warning alarm should be let off.Example 1: ((NOT A) OR(A AND B)) OR (NOT(A AND B))Example 2: ((A OR C) NOR(A AND B)) NAND (B XOR C)Practice Problem - Real Life Example:If traffic light A is green (on) traffic light, traffic light C can/must be green too. However, traffic light B can not be green then. Thus, if traffic light B is not on, neither is traffic light D.Hint:Created By: Lucie Charlotte Magister

Last update: 20/11/2014

Sources: