What are Armstrong axioms in DBMS?
Did you know?
Axioms in Armstrong are helpful in determining the functional dependencies of a particular database!
This forms a significant concept that is often asked from the programmers or coders to test their basic knowledge.
Developed by William. W. Armstrong in the year 1974, these axioms formed the complete set of rules that you have to follow in the database.
Just like the various concepts of DBMS like DBMS functionalities, its categories, anomalies in DBMS, Armstrong axioms are also crucial to be known by every user.
Are you intrigued to know about all these rules in depth?
Know in detail about the armstrong axioms in dbms and fill your knowledge bank with the right information!
So without further ado, let’s get started!
Armstrong axioms in DBMS
Let’s first get into the definition of this term. Armstrong’s axioms are touted as a mathematical notation which is helpful to find functional dependencies in a particular database.
This is defined as the list of rules or axioms of inferences which can be easily implemented on any type of relatable database. This is denoted generally with the symbol F++
The axiom is mainly useful to analyse, refine or maintain the database relation by applying three major types of modes which are:
- Axioms of Reflectivity: If the user name [ A] and the title [ B] is representing the name of a person. The relation between the A or B will hold least importance
- Axioms of Argumentation: If the user ID defines the name of a person, the user ID with an email quota will define the name of a person with its email quota
- Axioms of transitivity: If any user ID will determine the name of a person, and the person here is defining the department, the department will be defined by its user ID
Armstrong relation
An armstrong relation will be defined as the relation that will satisfy all the functional dependencies in their closure of F++and the only dependency related to the given set of the functional dependencies.
Unfortunately, for the given set of your dependencies, the minimum-size available for an armstrong relation may have a size with that of the number of various attributes in any dependency under consideration.
The rule of Armstrong relation
A set of F functional dependencies are given to you. Here, the Armstrong relation will satisfy all kinds of functional dependencies with the closure of F++ and to those of dependencies.
Though, in traditional paradigm designing of the database scheme will be given by the database designer. It also determines the set of various attributes and the set of data dependencies with all the attributes. Though, this type of information may be given as an input with the help of an algorithm in order to produce the database with the scheme that will satisfy the database condition by choosing a set of properties.
Armstrong axioms in DBMS Rules
There are basically two different types of rules in case of Armstrong axioms in DBMS. These can be in the form of primary rules and secondary rules.
Primary rules
In primary rules, R [U] will be the relation scheme over its set of attributes U. We will be denoting letters X,Y, Z alongside the subset of U. Union of two attributes will be X and Y. though, the standard of database theory will deal with certain sets of attributes.
Axiom of reflexivity
If A is denoted as the set of attributes and the B will be its subset. A holds B. In the case, A -B it will be trivial.
Axiom of Augmentation
In this case, if A → B, Y would be the attribute set. It holds AY →BY. It adds attributes in case of dependency. If A →B , AC →BC for the C
Secondary Rules
These rules are defined as the additional rules which includes decomposition, composition and the union which is described as follows:
Decomposition
If the X → Y, X → Y and X → Z
The proof of this would be:
X → YZ [ is given]
YZ → Y [ It is reflexivity]
X → Y [ It will be the transitivity of both 1 and 2 ]
Composition
If the X → Y, A → B then XA → YB
The proof of this would be
X → Y [ It is given]
A → B [ It is given]
XY – YA [ It is the augmentation of 1 and A]
XA → Y [ It is the decomposition of 3]
XA → XB [ It is the augmentation of 2 and x]
XA → B [ It is the decomposition of 5]
XA → YB [ It is the union of both 4 and 6]
Union or notation
If X → Y , X → Z, X → YZ
The proof of this would be
If X → Y [ is given]
X → Z [ is given]
X → XZ [ Augmentation of both 2 and X]
XZ → YZ [ Augmentation of both 1 and Z]
X → YZ [ transitivity of both 3 and 4]
Pseudocode
If X →Y , YZ →W, X Z → W
Here the self determination would be
I → I. This may directly follow from the axiom of reflexivity.
Extensivity
It is a special case augmentation when Z = X
If X → Y, X → XY
It can replace the augmentation as the axiom in a sense of augmentation that you can get from it together.
The proof of it would be
XZ → X [ This is reflexivity]
X → Y [ It is given]
X → Y [ It is transitivity of 1 and 2]
XZ → Y [ Transitivity of both 1 and 2]
XZ → XYZ [ Extensivity of the number 3]
XYZ → YZ [ This is reflexivity]
XZ → YZ [ This is transitivity of 4 and 5]
Wrapping up
DBMS is a vast field with so many related concepts like anomalies in DBMS and other data related concepts.
In this blog, we have discussed the armstrong axioms in dbms which is a crucial concept in DBMS. These inference rules should be known by every programmer for the better understanding.
Keep this guide as your mentor and make the most out of it!
