Postulate is a statement that is considered to be true without requirement of any proof. So, basically they are the assumptions which assumes that the proposition of the statement is true without any evidence produced. It helps in solving any problems as it forms the basis of deriving other logical statement and they are the fundamental propositions for proving the statement to be known as theorem. Postulate can be used to evident other theorems after one theorem has been proven. In this manner, few postulates can help in building the entire branch of mathematics. Postulates is also known by other name called axioms and such axioms applies to all the mathematic branch as they implies the assumption. They are considered to be assumption for any particular branch of mathematics and to any specific theory.
A classic example is provided by Euclidean theory and there are five common notions and five postulates that forms the basis of geometry of Euclid. They are regarded as the basic structure that helps in deriving theorems and lemmas. There are five postulates that forms the basis of the whole of Euclidean geometry and such postulates are known by the name of postulates of Euclid with the assumptions of postulates that are particular to geometry. The axioms are also known as the common notions. Euclid five postulates are considered to be assumptions about angles, lines and specific geometric concepts. Such postulates are listed below:
Euclid Common notions have application in all the mathematic branch and those common notion are listed below:
However, mathematicians can assume any number of postulates when developing the mathematical system using logical deductive reasoning. Theorem becomes equal to postulates when theorem is proved using the postulates. Therefore, total number of postulates are sought by the mathematicians for basing their reasoning presented. It is believed by the mathematician that the logical deduction of fifth postulate can be done from the other four postulates. It was in the nineteenth century when the mathematicians recognized that the logical consistency in the geometry is resulted by the five postulates and there would be other consistent geometries when fifth postulates is replaced by different assumptions. This can be explained with the help of an example.
The congruency of triangle can be proved using different postulates that are regarded as the rule of congruency such as SAS, ASA, SSS and AAS rule.
The postulates of SSS is used to prove that these two triangles are congruent. This is so because there is equality between the corresponding sides of both the triangles that is all the sides of the triangles are equal.
The language of arithmetic was formalized by the basic concepts determined by the figure of postulates and these basic concept include number, zero and the relationship presenting “successor of “. Furthermore, the fundamental laws of arithmetic was derived by the Italian mathematician based on the five postulates or axioms obeyed by the identified three concepts. Not only th formal foundation of arithmetic was provided by the postulates, but also forms the basis of construction of algebra. Nonetheless, these five postulates that assisted in deriving the arithmetic laws are different and they comprised of:
The limitations of the proof provided by the postulates was evident from the significant discoveries in the logical and mathematical field and thereby invalidated by the axioms of
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