Download ASSIGNMENT 3

ASSIGNMENT 3 – AXIOMATIC SYSTEMS
1. Undefined Terms: Undefined terms are those terms that are accepted without any further definition. For
example, in modern Euclidean geometry the terms ‘point and ‘line’ are typically left undefined.
2. Axioms (or Postulates): An axiom (or postulate) is a logical statement about terms that is accepted
without proof. For example, the statement “A straight line can be drawn from any point to any point” is the
first of Euclid’s five postulates.
3. Defined terms: From a base of undefined terms and accepted axioms, we define other terms.
4. Theorems: Defined terms and accepted axioms are used to argue the truth of other statements call
theorems.
Axiomatic System: A system comprising of the four components above, along with some basic rules of logic,
is an axiomatic system.
Example 1: A game of chess




Playing pieces and chess board – undefined terms
The rules - axioms
A particular playing piece (say bishop) – a defined term
A particular configuration of the game, for example one player holding the another in check – a theorem
Example 2:
a) Students and classes – undefined terms
b) Axioms:
a) There are exactly 3 students
b) For each pair of students, there is exactly one class in which they are enrolled
c) Not all students belong to one class
d) Two separate classes share at least one student in common
c) Theorems
Deductions from this set of axioms:
Theorem 1: Two separate classes share one and only one student in common
Theorem 2: There are exactly 3 classes in our system
Theorem 3: Each class has exactly two students
Exercise: Prove the above theorems.
Example 3:


Students in a classroom, different flavors of ice cream
Axioms
a) There are exactly five flavors of ice cream
b) Given any two different flavors, there is exactly one child who likes these flavors
c) Every child likes exactly two different flavors among the five
d) Theorems in exercises below:
1. How many children are there in the classroom?
2. Show any pair of children likes at most one common flavor
3. Show that for each flavor there are exactly four children who like that flavor.
Example 4
Group: A group is defined as a set G of undefined objects called elements and a binary operation “  ” that
relates two objects to a third. The axioms for a group are
a) For all elements x and y, the binary operation on x, y is again an element of G. i.e. for x, y  G,
x  y G .
b) For all x, y, z  G, ( x  y )  z  x  ( y  z ) , i.e. the binary operation is associative
c) There is a special element e  G such that x  e  x for all x  G . e is called the identity
element of G.
d) Given x  G , there is an element x 1  G , such that x  x 1  e . The element x 1 is called the
inverse of x.
Exercises
1. Show that if x, y, z  G , and x  z  y  z , then x = y.
2. Show that x  e  e  x .
3. Show that a group can have only one identity