Algebraic System
J.M.Basilla
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2011 Math 1
JB
Tic-tac-toe
Group Theory
Julius Magalona Basilla
Institute of Mathematics
University of the Philippines-Diliman
[email protected]
Game: an
axiomatic system
Axioms of Set
Theory
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Math 1
General Education Mathematics
Lecture
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Algebraic Systems: Groups and Rings
Modular Addition
Games: axiomatic systems
Algebraic System
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J.M.Basilla
Games are best examples of axiomatic systems
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Group Theory
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Creating a new game is like creating a new axiomatic
system
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Tic-tac-toe
Axioms of Set
Theory
Chess, Tic-tac-toe, dama(checkers),
games-of-the-generals, chinese checkers binggo
chess
JB
Game: an
axiomatic system
Modular Addition
The game of Tic-tac-toe
Algebraic System
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J.M.Basilla
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Group Theory
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The player who successfully places three marks in a
line wins the game.
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Tic-tac-toe
Axioms of Set
Theory
A two-person game , X and O, who takes turn
marking the spaces in a 3 × 3 grid.
The X player goes first.
Game: an
axiomatic system
Modular Addition
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O X X
X O O
X O X
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Sample- tictac-toe game
Algebraic System
J.M.Basilla
Game: an
axiomatic system
Axioms of Set
Theory
Tic-tac-toe
Group Theory
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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J.M.Basilla
Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
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Group Theory
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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J.M.Basilla
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Group Theory
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The first player wins the
game
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Tic-tac-toe
Axioms of Set
Theory
X
JB
Game: an
axiomatic system
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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Tic-tac-toe
Group Theory
The first player wins the
game
JB
Game: an
axiomatic system
Axioms of Set
Theory
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J.M.Basilla
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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Tic-tac-toe
Group Theory
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The first player wins the
game
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Game: an
axiomatic system
Axioms of Set
Theory
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J.M.Basilla
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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Tic-tac-toe
Group Theory
X
The first player wins the
game
JB
Game: an
axiomatic system
Axioms of Set
Theory
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O O
X
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J.M.Basilla
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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J.M.Basilla
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X O O
X
X
The first player wins the
game
Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
Group Theory
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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J.M.Basilla
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X O O
X
X
O
The first player wins the
game
Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
Group Theory
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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J.M.Basilla
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X O O
X X
X
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The first player wins the
game
Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
Group Theory
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
J.M.Basilla
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X O O
X X
X
O
X player wins the game.
The first player wins the
game
Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
Group Theory
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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X O O
O X
X
The first player wins the
game
Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
Group Theory
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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J.M.Basilla
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X O O
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The first player wins the
game
Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
Group Theory
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
J.M.Basilla
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X O O
O X
X
X
X player wins the game.
The first player wins the
game
Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
Group Theory
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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J.M.Basilla
Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
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Group Theory
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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J.M.Basilla
Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
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The game ends in a
draw.
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Group Theory
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
X
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Tic-tac-toe
Group Theory
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Game: an
axiomatic system
Axioms of Set
Theory
The game ends in a
draw.
JB
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J.M.Basilla
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
X
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Tic-tac-toe
Group Theory
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Game: an
axiomatic system
Axioms of Set
Theory
The game ends in a
draw.
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J.M.Basilla
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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Tic-tac-toe
Group Theory
The game ends in a
draw.
JB
Game: an
axiomatic system
Axioms of Set
Theory
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J.M.Basilla
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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Tic-tac-toe
Group Theory
The game ends in a
draw.
JB
Game: an
axiomatic system
Axioms of Set
Theory
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X X O
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J.M.Basilla
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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Tic-tac-toe
Group Theory
The game ends in a
draw.
JB
Game: an
axiomatic system
Axioms of Set
Theory
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J.M.Basilla
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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Tic-tac-toe
Group Theory
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Game: an
axiomatic system
Axioms of Set
Theory
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O
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X X O
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The game ends in a
draw.
JB
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J.M.Basilla
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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Tic-tac-toe
Group Theory
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Game: an
axiomatic system
Axioms of Set
Theory
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O
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X X O
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The game ends in a
draw.
JB
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J.M.Basilla
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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Tic-tac-toe
Group Theory
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Game: an
axiomatic system
Axioms of Set
Theory
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O X X
X X O
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The game ends in a
draw.
JB
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J.M.Basilla
Modular Addition
Tic tac toe Theorem
The first player always wins the game or draw
Algebraic System
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Tic-tac-toe
Group Theory
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Game: an
axiomatic system
Axioms of Set
Theory
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O X X
X X O
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The game ends in a
draw.
JB
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J.M.Basilla
Modular Addition
Axioms of Set Theory
Algebraic System
Primitive terms : elements, sets, universal set.
Axioms
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elements are members of the universal set.
A collection of some(perhaps all) of the elements
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J.M.Basilla
Some Definitions
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If every element of A is also an element of B, we call
A a subset of B. This idea is conveyed in symbol as
A ⊂ B.
Two sets A and B are said to be equal if and only if
A ⊂ B and B ⊂ A.
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If A ⊂ B and B ⊂ C then A ⊂ C.
The complement of A ∩ B is the union of the
complements of A and B.
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Some theorems
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’ denote the ”complement of”.
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(A ∩ B)0 = A0 ∪ B 0 ,
Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
Group Theory
Modular Addition
Modular Addition
Algebraic System
Modular Addition
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For two numbers a and b, we define a ⊕n b to be the
remainder when a + b is divided by n.
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J.M.Basilla
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Group Theory
2. 15 ⊕26 12 = 1
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Tic-tac-toe
Axioms of Set
Theory
1. 5 ⊕10 7 = 2
3. 4 ⊕11 7 = 0
Game: an
axiomatic system
Modular Addition
Modular Addition
Algebraic System
1. For any numbers a and b, a ⊕n b = b ⊕n a.
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2. For any numbers a, b and c,
a ⊕n (b ⊕n c) = (a ⊕n b) ⊕n c.
J.M.Basilla
3. For any numbers a , a ⊕n 0 = a.
Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
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Group Theory
Modular Addition
Modular Addition
Algebraic System
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4. For each number a, we can find exactly one number
b such that a ⊕n b = 0. We call b, the negative of a
modulo n
Since
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−7 = 4.
and
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−4 = 7,
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⊕4 0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 0 1
3 3 0 1 2
5. In dealing with ⊕n , it remains for us to consider only
the numbers 0, 1, . . ., n − 1.
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Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
Group Theory
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4 ⊕11 +7 = 0,
J.M.Basilla
Modular Addition
Addition Tables modulo n
Algebraic System
Addition Table modulo 2
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Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
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= 0 ⊕2 1 ⊕2 0
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Group Theory
0 ⊕2 (1 ⊕2 1) ⊕2 1 ⊕2 0 = 0 ⊕2 0 ⊕2 1 ⊕2 0
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= 1.
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= 1 ⊕2 0
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1
J.M.Basilla
Modular Addition
Addition Tables modulo n
Algebraic System
Addition Table modulo 3
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⊕3
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J.M.Basilla
Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
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Addition Table modulo 4
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Group Theory
Modular Addition
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⊕26
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1 7
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⊕26
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⊕26
2 2
1 4
1 0
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1 9
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Examples
Algebraic System
J.M.Basilla
Game: an
axiomatic system
Axioms of Set
Theory
Tic-tac-toe
Group Theory
Modular Addition
Axioms of Group Theory
Algebraic System
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a ∗ b is an element of G whenever a and b are in G.
Associative Property holds. (i.e.
(∀(a, b, c) ∈ G3 )(a ∗ (b ∗ c) = (a ∗ b) ∗ c).)
Hence, a ∗ b ∗ c := a ∗ (b ∗ c) = (a ∗ b) ∗ c.
The exists an element e, called the identity such that
a ∗ e = e ∗ a for all a ∈ G.
Every element has an inverse. That is for every
a ∈ G, there exists an element a−1 such that
a ∗ a−1 = e
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A nonempty set G.
A composition rule ∗ on G such that
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J.M.Basilla
Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
Group Theory
Modular Addition
A theorem
Algebraic System
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The identity element e is unique.
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J.M.Basilla
Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
Proof.
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Group Theory
Suppose there are 2.
Call them e1 and e2 .
e1 = e1 ∗ e2 , since e2 is an identity element.
e2 = e1 ∗ e2 , since e1 is an identity element.
Therefore, e1 = e2 .
Modular Addition
Another theorem
Algebraic System
The inverse of each element is unique.
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Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
Proof.
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Group Theory
Suppose there are 2.
Call them b and c.
We have a ∗ b = b ∗ a = e and a ∗ c = c ∗ a = e.
b=b∗e=b∗a∗c=e∗c=c
Therefore, b = c.
Modular Addition
Consistency Models
examples of groups
Algebraic System
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1. Take G = {0, 1} and define a + b to be the addition
modulo 2, given by the table.
2 2
⊕26 1 4
1 0
2. Take G = {0, 1, 2} and define a + b to be the addition
modulo 3, given by the table.
⊕3 0 1 2
0 0 1 2
1 1 2 0
2 2 0 1
3. In general if G = {0, 1, 2, . . . , n} and define a + b to
be the addition modulo n, we get a group.
J.M.Basilla
Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
Group Theory
Modular Addition
Modular Multiplication
Algebraic System
Modular Multiplication
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For two numbers a and b, we define a ⊗n b to be the
remainder when ab is divided by n.
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Group Theory
2. 3 ⊗26 12 = 10
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Tic-tac-toe
Axioms of Set
Theory
1. 5 ⊗10 7 = 5
3. 4 ⊗11 7 = 6
Game: an
axiomatic system
Modular Addition
Modular Multiplication
Algebraic System
1. For any numbers a and b, a ⊗n b = b ⊗n a.
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2. For any numbers a, b and c,
a ⊗n (b ⊗n c) = (a ⊗n b) ⊗n c.
J.M.Basilla
3. For any numbers a , a ⊗n 1 = a.
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5. In dealing with ⊗n , it remains for us to consider only
the numbers 0, 1, . . ., n − 1.
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6. The set {0, 1, . . . , n − 1} does not form a group under
the operation ⊗n unless n is a prime number.
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Tic-tac-toe
Axioms of Set
Theory
4. For each number a, we may not find a number b such
that a ⊗n b = 1. However, if there is such we call b,
the inverse of a modulo n
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Game: an
axiomatic system
Group Theory
Modular Addition
Multiplication Tables modulo n
Algebraic System
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Game: an
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Tic-tac-toe
Axioms of Set
Theory
= 0 ⊗2 1 ⊗2 0
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0 ⊗2 (1 ⊗2 1) ⊗2 1 ⊗2 0 = 0 ⊗2 1 ⊗2 1 ⊗2 0
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Group Theory
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=0
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= 0 ⊗2 0
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Multiplication Table modulo 2
Modular Addition
Multiplication Tables modulo n
Algebraic System
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Game: an
axiomatic system
Tic-tac-toe
Axioms of Set
Theory
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Xor
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⊗4
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Modular Addition
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Multiplication Table modulo 4
⊗2
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⊕2
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Group Theory
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J.M.Basilla
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Multiplication Table modulo 3
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Rings
Algebraic System
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1. a collection of objects subject to two operations,
often called multiplication and addition, satisfying the
conditions we had for modulo addition and modulo
multiplication are called rings.
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Tic-tac-toe
Group Theory
3. Theoretical Mathematicians are concerned with
investigating and determining properties of a general
ring and their consequences wile those who are in
the applied disciplines tries to make use of specific
rings.
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Game: an
axiomatic system
Axioms of Set
Theory
2. These objects are studied deeper in areas of
Mathematics such as Ring Theory, Commutative
Algebra, Algebraic Geometry and in Algebraic
Number Theory.
4. In Cryptography, Coding Theory and Information
sciences, the finite rings are well used.
J.M.Basilla
Modular Addition