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Boolean Algebra

Background

Boolean algebra is a form of algebra where the variables only have the values true and false; 1 and 0, respectively.

The operations of Boolean algebra are:

 

• 

AND operation (denoted AB),

• 

OR operation (denoted A+B),

• 

NOT operation (denoted A')

 

 These operations are defined as follows:

 

NOT

AND

OR

0' = 1

1' = 0

0  0 = 0

0  1 = 0

1  0 = 0

1  1 = 1

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 1

 

You will notice that these are the basic logic gates described in the Logic Gates Math App. Boolean algebra is fundamental for digital logic.

Laws of Boolean Algebra

Note: Every law in Boolean algebra has two forms that are obtained by exchanging all the ANDs to ORs and 1s to 0s and vice versa. This is known as the Boolean algebra duality principle.  The order of operations for Boolean algebra, from highest to lowest priority is NOT, then AND, then OR. Expressions inside brackets are always evaluated first.

 

1. Commutative Law

(a) A + B = B + A

(b) A  B = B  A

 

2. Associative Law

(a) (A + B) + C = A + (B + C)

(b) (A  B)  C = A  (B  C)

 

3. Distributive Law

(a) A  (B + C) = A  B + A  C

(b) A + (B  C) = (A + B)  (A + C)

 

4. Null Law

(a) 1 + A = 1   

(b) 0  A = 0

 

5. Identity law

(a) 0 + A = A

(b) 1  A = A

 

6. Idempotent Law

(a) A + A = A

(b) A  A = A

 

7. Complementarity Law

(a) A' + A = 1

(b) A'  A = 0

 

8. Uniting Theorem

(a) A  B + A  B' = A

(b) (A + B)  (A + B') = A

 

9. Absorption Theorem

(a) A + A  B = A

(b) A  (A + B) = A

 

10. Adsorption Theorem

(a)  A + A'  B = A + B

(b) A  (A' + B) = A  B

 

11. De Morgan's Theorem

(a) (A + B)' = A'  B'

(b) (A  B)' = A' + B'

The following activity allows you to practice remembering the basic laws of Boolean algebra and simplifying Boolean expressions. The objective of the game is to complete the equations below in a timely manner.  

To play, use the laws (above) to solve for the missing variable(s) and/or operator(s) in the equation. The orange square indicates a blank in the equation. Before the mouse reaches the gap, your objective is to use the options (variables and operators) below to correctly complete the equation. With the correct response in place, the mouse is able to run over the gap and obtain the cheese. If the response is incorrect, the mouse falls through. The difficulty (easy, medium, hard) allows you to adjust the speed of the mouse.

 

CHEESE HUNT

A Boolean Algebra Game

 

Choose Difficulty:

 

Options:

 

 

 

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