Why Does a' * a' = a'? Explained!

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SUMMARY

The equation a' * a' = a' is a direct result of the idempotent law in Boolean algebra, which states that any variable ANDed with itself yields the same variable. In this case, a' represents the complement of a, and thus a' * a' simplifies to a'. Additionally, the expression (a')' equals a, demonstrating the involution property of complements in Boolean algebra. A truth table confirms that for binary values, the product of a variable with itself always returns the original variable.

PREREQUISITES
  • Understanding of Boolean algebra principles
  • Familiarity with the concepts of complements and idempotent laws
  • Ability to construct and interpret truth tables
  • Knowledge of basic logical operations (AND, OR, NOT)
NEXT STEPS
  • Study the idempotent law in Boolean algebra
  • Learn about the involution property of complements
  • Practice creating truth tables for various Boolean expressions
  • Explore applications of Boolean algebra in digital circuit design
USEFUL FOR

This discussion is beneficial for students of computer science, electrical engineers, and anyone involved in digital logic design or Boolean algebra applications.

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why does a' * a' = a' ??

this isn't a homework problem, but i am just wondering why because i used this to simplify another boolean equation. oh, and what's the difference between a' * a' and (a')' ? how do you even get (a')'?? confused :\
 
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Every element in a Boolean algebra is idempotent: xx = x for every x.

a'a' is the product of a' with itself; (a')' = a is a complemented twice. Complement twice is the identity (complement is an involution).
 
Make a truth table to see why.
X can equal only 1 or 0 thus the truth table for x*x is below :
--------
x x x*x
0 0 0
1 1 1
-------

As you see if X = 0, then X*X = 0 = X, and if X = 1, then X*X = 1 = X. Thus X*X = X
 

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