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Description
Boolean Postulate, Properties, and Theorems
The following postulate, properties, and theorems are valid in Boolean Algebra and are used in simplification of logical expressions or functions:
POSTULATES are self - evident truths.
1a: $A=1$ (if A ≠ 0) 1b: $A=0$ (if A ≠ 1)
2a: $0∙0=0$ 2b: $0+0=0$
3a: $1∙1=1$ 3b: $1+1=1$
4a: $1∙0=0$ 4b: $1+0=1$
5a: $\overline{1}=0$ 5b: $\overline{0}=1$
PROPERTIES that are valid in Boolean Algebra are similar to the ones in ordinary algebra
Commutative $A∙B=B∙A$ $A+B=B+A$
Associative $A∙(B∙C)=(A∙B)∙C$ $A+(B+C)=(A+B)+C$
Distributive $A∙(B+C)=A∙B+A∙C$ $A+(B∙C)=(A+B)∙(A+C)$
THEOREMS that are defined in Boolean Algebra are the following:
1a: $A∙0=0$ 1b: $A+0=A$
2a: $A∙1=A$ 2b: $A+1=1$
3a: $A∙A=A$ 3b: $A+A=A$
4a: $A∙\overline{A}=0$ 4b: $A+\overline{A}=1$
5a: $\overline{\overline{A}}=A$ 5b: $A=\overline{\overline{A}}$
6a: $\overline{A∙B}=\overline{A}+\overline{B}$ 6b: $\overline{A+B}=\overline{A}∙\overline{B}$
By applying Boolean postulates, properties and/or theorems we can simplify complex Boolean expressions and build a smaller logic block diagram (less expensive circuit).
For example, to simplify $AB(A+C)$ we have:
$AB(A+C)$ distributive law
=$ABA+ABC$ cumulative law
=$AAB+ABC$ theorem 3a
=$AB+ABC$ distributive law
=$AB(1+C)$ theorem 2b
=$AB1$ theorem 2a
=$AB$
Although the above is all you need to simplify a Boolean equation. You can use an extension of the theorems/laws to make it easier to simplify. The following will reduce the amount of steps required to simplify but will be more difficult to identify.
7a: $A∙(A+B)=A$ 7b: $A+A∙B=A$
8a: $(A+B)∙(A+\overline{B})=A$ 8b: $A∙B+A∙\overline{B}=A$
9a: $(A+\overline{B})∙B=A∙B$ 9b: $A∙\overline{B}+B=A+B$
10: $A⊕B=\overline{A}∙B+A∙\overline{B}$
11: $A⊙B=\overline{A}∙\overline{B}+A∙B$
⊕ = XOR, ⊙ = XNOR
Now using these new theorems/laws we can simplify the previous expression like this.
To simplify $AB(A+C)$ we have:
$AB(A+C)$ distributive law
=$ABA+ABC$ cumulative law
=$AAB+ABC$ theorem 3a
=$AB+ABC$ theorem 7b
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- Version: 1.0
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Android Apps›Education - OS:
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22 Mb - Content Rating:
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«Boolean simplifier» is a Education app for Android, developed by «sajith tiyenshan». It was first released on and last updated on . This app is currently free. This app has not yet received any ratings or reviews on AppAgg. AppAgg continuously tracks the price history, ratings, and user feedback for «Boolean simplifier». Subscribe to this app or follow its RSS feed to get notified about future discounts or updates. AppAgg does not host applications or distribute software. All trademarks, logos and screenshots belong to their respective owners.
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