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Axiomatic System
J.M.Basilla
Axiomatic System
Math 1
General Education Mathematics
Lecture
Julius Magalona Basilla
Institute of Mathematics
University of the Philippines-Diliman
[email protected]
2011 Math 1
Axiomatic System
The elements of an
axiomatic System
Axiomatic System
Axiomatic System
J.M.Basilla
An axiomatic system or a postulate system consists
of some undefined terms and a list of statements,
called axioms or postulate, concerning the primitive
terms.
A mathematical theory is obtained from an axiomatic
system by proving new statements called theorems
using only the axiom and/or previously proved
theorems.
Definitions are made in the process in order to be
more concise and to facilitate discussions.
Axiomatic System
The elements of an
axiomatic System
The elements of an Axiomatic System
Primitive Terms
Axiomatic System
J.M.Basilla
Axiomatic System
The elements of an
axiomatic System
Primitive terms are undefined terms/objects.
Primitive terms form the very foundation of an
axiomatic system.
The behavior and properties of primitive terms are
understood through the axioms of an axiomatic
system.
The elements of an Axiomatic System
Axioms
Axiomatic System
J.M.Basilla
Axiomatic System
The elements of an
axiomatic System
Axioms are rules/description of primitive terms which
are accepted as truth.
It is through these axioms that we come to
understood a primitive term.
Any object that can be shown to satisfy the axioms
can be considered as an interpretation of the
primitive term.
Chess
Primitive terms in the game ”Chess”
person(playing)
knight
chessboard
rook/castle/tower
pawn
queen
bishop
king
Some axioms
There are exactly two persons playing in a game of
chess.
The board has sixty-four squares of alternating
colors.
Each player has a set of sixteen pieces with the
following composition
Pawns 8 Bishops 2 Knights 2
Rooks 2
Queen 1
King
1
Axiomatic System
J.M.Basilla
Axiomatic System
The elements of an
axiomatic System
Newtonian Physcis
some primitive terms in Newtonian Physics
matter energy mass
Some axioms: Newtons law of motions
First law
A body persists its state of rest or of uniform
motion unless acted upon by an external
unbalanced force."
Axiomatic System
J.M.Basilla
Axiomatic System
The elements of an
axiomatic System
Newtonian Physcis
Axiomatic System
Second law
Force equals mass times acceleration (F = ma)":
the net force on an object is equal to the mass of
the object multiplied by its acceleration.
Third law
To every action there is an equal and opposite
reaction.
J.M.Basilla
Axiomatic System
The elements of an
axiomatic System
Set Theory
A mathematical example
Axiomatic System
J.M.Basilla
Axiomatic System
The elements of an
axiomatic System
Primitive Terms : elements and sets
Axioms: An element is a member of a set. If this is
the case, we say the element x is and element of the
set S, written x ∈ S.
Definitions
As an axiomatic system becomes more and more
complicated, definitions are introduced to facilitate
discussions and analyses.
For example, in a game of chess, some of the
defined terms are
check
mate
stalemate
en passant castling
draw
In set theory, there is a definition, A set A is a subset
of B if and only if every element of A is also an
element of B.
Axiomatic System
J.M.Basilla
Axiomatic System
The elements of an
axiomatic System
Theorems and Propositions
Axiomatic System
J.M.Basilla
1. A statement which has been proven on the basis of
previously proven and/or previously accepted true
statements suchs as other theorems and/or axioms.
2. Often a theorem is stated with two parts; the
hypothesis and the conclusion. The hypothesis is a
condition that when satisfied results to the
occurrence/validity of the conclusion.
3. The derivation of a theorem is the proof of truth of the
resulting statements once the hypothesis is satisfied.
4. Well known theorems are Four-color theorem;
Fermat(-Wiles) theorem, Pythagorean Theorem
Axiomatic System
The elements of an
axiomatic System