4.3.8

Boolean Rules

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Rules of Boolean Operations

There are a number of rules that can be used to simplify boolean expressions. Simplified expressions are cheaper and easier to implement as fewer logic gates are needed.

Absorption

Absorption

  • We can simplify expressions in the following way:
    • X+(X.Y)=X+X=XX+(X.Y) = X +X = X
  • We can only do this if:
    • The value outside the brackets must be present within the brackets.
    • The operators inside and outside the brackets must be different.
  • We say that Y has been absorbed.
Commutative

Commutative

  • The commutative rule states that values in AND or OR expressions can be swapped without changing the result:
    • X.Y=Y.XX. Y = Y . X
    • X+Y=Y+XX + Y = Y + X
Distributive

Distributive

  • A value outside of a set of brackets can be distributed between the values within the brackets:
    • X.(Y+Z)=(X.Y)+(X.Z)X . (Y + Z) = (X . Y) + (X . Z)
    • X+(Y.Z)=(X+Y).(X+Z)X + (Y . Z) = (X + Y) .(X + Z)
Associative

Associative

  • The associative rule confirms that the order of operators in an expression has no impact on the result:
    • (X+Y)+Z=X+(Y+Z)(X + Y) + Z = X + (Y + Z)
    • (X.Y).Z=X.(Y.Z)(X . Y) .Z= X . (Y . Z)
Double Negation

Double Negation

  • The equivalent of any double-negative is a positive:
    • (X‾)‾=X\overline{(\overline{X})} = X
De Morgan's Law

De Morgan's Law

  • De Morgan's law is as follows:
    • X+Y‾=X‾.Y‾\overline{X + Y} = \overline{X} . \overline{Y}
    • X.Y‾=X‾+Y‾\overline{X . Y} = \overline{X} + \overline{Y}
  • We can use this law to convert part of an expression so the whole expression only uses one type of operator.
  • This reduces the cost as only one type of gate is required.
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Compression, Encryption & Hashing

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Data Types

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4.3

Boolean Algebra

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6

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Thinking Ahead

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6.3

Thinking Procedurally

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Algorithms

8.1

Algorithms

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Practice questions on Boolean Rules

Can you answer these? Test yourself with free interactive practice on Seneca — used by over 10 million students.

  1. 1
    Which law allows us to simplify the expression shown: <br>$$X+(X.Y) = X +X$$Multiple choice
  2. 2
    Conditions for absorption rule:Fill in the list
  3. 3
    Use the distributive rule to simplify: <br>$$A . (B + C)$$Multiple choice
  4. 4
    Which rule confirms that the order of operators in an expression has no impact on the result?Multiple choice
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Boolean Identities

Flip Flops