哥伦比亚大学-离散数学-笔记-第5-8章-3
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n
det(A) =
σ ∈Sn
(−1)#inv
i=1
aσ(i),i
(19)
This function is O(n!) as described above. How can we solve systems of equations efficiently? Can we use this idea to determine whether solutions exist to other algebraic problems (not just systems of equations)? Next time: Gaussian elimination, upper triangular form
Discrete Math
Lecture 13
Page 4 of 4
Note: f◦(g◦h) = (f◦g)◦h f is perm of A, f-1 is its inverse f-1 ◦f = e, f ◦f-1 = e We discussed operation of function composition which takes two functions and finds their composite. We saw the set of all n! permutations, Sn (the symmetric group) which was determined by n. Example remember last time, 3 people sit in chairs, then S3 contains the six permutations: = 1 1 1 2 2 2 2 1 3 3 3 3 α= 1 1 1 2 2 3 2 3 3 2 3 1 β= 1 3 1 3 2 1 2 2 3 2 3 1
Discrete Math
Lecture 13
scribe: Alexandra Taylor-Gutt
Last time: permutations (bijections), combinatorics Algebra is more than the science of solving equations. In the 1800’s mathematicians discovered different types of algebras that involved a set, together with a rule for combining two elements of that set. Examples: -set of colors, operation of mixing two -set of musical sounds/notes, operation of combining sounds Boolean Algebra is foundation of Computer Science used to design logic gates and circuits: Write A+B for A ∪ B A • B for A ∩ B A+B=B+A A• B = B • A A• (B+C) = AB + AC A+∅=A A•∅=∅ Our study of algebra must include these new algebras and not be limited to solving equations. This motivates the idea of a group as the simplest algebraic structure. Groups are used to study symmetry in many forms, including things that, at first, may not seem to be related to mathematics. Definition A group (G, ∗) is a set G together with an operation ∗ that preserves the four axioms: 1. Closure 2. Associativity 3. Identity: ∃e∈G, s.t. e∗a = a and a∗e = a ∀a∈G 4. Inverse ∀a∈G ∃a-1 s.t. aa-1 =e and a-1 a=e Examples (Z, +) (Q* ,×) (Q, +) (R*,×) Check that these satisfy the four axioms above. Finite Groups Ex: Group of integers modulo n, under the operation of addition mod n called (Zn , +) or Zn , where Zn = {0, 1, 2, ..., n − 1}. We can also define the group (Zn , ×) of integers modulo n under multiplication. Let’s look at the operation tables for these finite groups:
Permutations can be written as a product of disjoint cycles Question: When does any transitive subgroup of permutations contain a cyclic shift? We defined Leibniz Formula for the determinant using permutations:
γ=
δ=
κ=
To find α ◦ β we do... (α ◦ β )(1) = α(β (1)) = α(3) = 2 (α ◦ β )(2) = α(β (2)) = α(1) = 1 (α ◦ β )(3) = α(β (3)) = α(3) = 3 (15) (16) (17) (18) Therefore (α ◦ β ) = γ . We could make a table of this: ◦ α β γ δ κ α β γ δ κ α α κ δ γ β β β γ δ κ α γ γ β α κ δ δ δ κ α β γ κ κ δ γ β α
Discrete Math
Lecture 13
wenku.baidu.com
Page 2 of 4
Figure 2: Operation tables for Z6 What is e2 ∗ e1 ? e2 ∗ e1 = e1 By Axiom 3, e1 = e2 . Question: Can a group have more than one inverse element (a)? Inverse Proof. What is a1 ∗ (a ∗ a2 )? a1 ∗ (a ∗ a2 ) = a1 ∗ e = a1 What is a2 ∗ (a ∗ a1 )? a2 ∗ (a ∗ a1 ) = a2 ∗ e = a2 By Axiom 2 and 4: a1 = a2 (7) (5) (6) (3) (4) (2)
Discrete Math
Lecture 13
Page 3 of 4
Figure 3: Subgroup
Figure 4: Cayley Diagram for the Example Below Example Consider an operation ”multiply by a” (or ”multiply by b”): e→ a→ a2 → a3 → and a set G with the rules a2 = e, b3 = e, and ba = ab2 : G = {e, a, b, b2 , ab, ab2 } Question Evaluate the expression: (ab) ∗ (ab2 ) = a ba b2 =a b = a
Definition Let S be a nonempty subset of G if S 1. closed under multiplication 2. closed with respect to inverse then S is a subgroup of G Example Even integers under addition (subgroup of integers under addition)
The End.
2 e ab2 2 4
(8)
(9) (10)
b
3
b
(11) (12) (13) (14)
e
= eeb = eb =b
A Cayley Diagram represents same info as group operation table, there is one point for each element of G, arrows represent effect of applying generator. Let’s finally talk about permutations We will see that the set of Permutations form a group under the operation of function composition. -inverse of any bijection exists ⇒ inverse permutation exists (as by definition a permutation is a bijection from a set onto itself). Cont.
Figure 1: Operation tables for Z5 Group need not be commutative (a commutative group is called Abelian after mathematician Niels Abel). Question: Can a group have more than one identity element (e)? Identity Proof. let e1 , e2 be two identity elements. What is e1 ∗ e2 ? e1 ∗ e2 = e2 (1)
Some groups have an interesting property: All elements obtained by repeatedly applying group operation to a particular element. This is called a cyclic group, element is called generator 1 = 1 mod n 1 + 1 = 2 mod n 1 + 1 + 1 = 3 mod n not every number can be the generator Cont.
det(A) =
σ ∈Sn
(−1)#inv
i=1
aσ(i),i
(19)
This function is O(n!) as described above. How can we solve systems of equations efficiently? Can we use this idea to determine whether solutions exist to other algebraic problems (not just systems of equations)? Next time: Gaussian elimination, upper triangular form
Discrete Math
Lecture 13
Page 4 of 4
Note: f◦(g◦h) = (f◦g)◦h f is perm of A, f-1 is its inverse f-1 ◦f = e, f ◦f-1 = e We discussed operation of function composition which takes two functions and finds their composite. We saw the set of all n! permutations, Sn (the symmetric group) which was determined by n. Example remember last time, 3 people sit in chairs, then S3 contains the six permutations: = 1 1 1 2 2 2 2 1 3 3 3 3 α= 1 1 1 2 2 3 2 3 3 2 3 1 β= 1 3 1 3 2 1 2 2 3 2 3 1
Discrete Math
Lecture 13
scribe: Alexandra Taylor-Gutt
Last time: permutations (bijections), combinatorics Algebra is more than the science of solving equations. In the 1800’s mathematicians discovered different types of algebras that involved a set, together with a rule for combining two elements of that set. Examples: -set of colors, operation of mixing two -set of musical sounds/notes, operation of combining sounds Boolean Algebra is foundation of Computer Science used to design logic gates and circuits: Write A+B for A ∪ B A • B for A ∩ B A+B=B+A A• B = B • A A• (B+C) = AB + AC A+∅=A A•∅=∅ Our study of algebra must include these new algebras and not be limited to solving equations. This motivates the idea of a group as the simplest algebraic structure. Groups are used to study symmetry in many forms, including things that, at first, may not seem to be related to mathematics. Definition A group (G, ∗) is a set G together with an operation ∗ that preserves the four axioms: 1. Closure 2. Associativity 3. Identity: ∃e∈G, s.t. e∗a = a and a∗e = a ∀a∈G 4. Inverse ∀a∈G ∃a-1 s.t. aa-1 =e and a-1 a=e Examples (Z, +) (Q* ,×) (Q, +) (R*,×) Check that these satisfy the four axioms above. Finite Groups Ex: Group of integers modulo n, under the operation of addition mod n called (Zn , +) or Zn , where Zn = {0, 1, 2, ..., n − 1}. We can also define the group (Zn , ×) of integers modulo n under multiplication. Let’s look at the operation tables for these finite groups:
Permutations can be written as a product of disjoint cycles Question: When does any transitive subgroup of permutations contain a cyclic shift? We defined Leibniz Formula for the determinant using permutations:
γ=
δ=
κ=
To find α ◦ β we do... (α ◦ β )(1) = α(β (1)) = α(3) = 2 (α ◦ β )(2) = α(β (2)) = α(1) = 1 (α ◦ β )(3) = α(β (3)) = α(3) = 3 (15) (16) (17) (18) Therefore (α ◦ β ) = γ . We could make a table of this: ◦ α β γ δ κ α β γ δ κ α α κ δ γ β β β γ δ κ α γ γ β α κ δ δ δ κ α β γ κ κ δ γ β α
Discrete Math
Lecture 13
wenku.baidu.com
Page 2 of 4
Figure 2: Operation tables for Z6 What is e2 ∗ e1 ? e2 ∗ e1 = e1 By Axiom 3, e1 = e2 . Question: Can a group have more than one inverse element (a)? Inverse Proof. What is a1 ∗ (a ∗ a2 )? a1 ∗ (a ∗ a2 ) = a1 ∗ e = a1 What is a2 ∗ (a ∗ a1 )? a2 ∗ (a ∗ a1 ) = a2 ∗ e = a2 By Axiom 2 and 4: a1 = a2 (7) (5) (6) (3) (4) (2)
Discrete Math
Lecture 13
Page 3 of 4
Figure 3: Subgroup
Figure 4: Cayley Diagram for the Example Below Example Consider an operation ”multiply by a” (or ”multiply by b”): e→ a→ a2 → a3 → and a set G with the rules a2 = e, b3 = e, and ba = ab2 : G = {e, a, b, b2 , ab, ab2 } Question Evaluate the expression: (ab) ∗ (ab2 ) = a ba b2 =a b = a
Definition Let S be a nonempty subset of G if S 1. closed under multiplication 2. closed with respect to inverse then S is a subgroup of G Example Even integers under addition (subgroup of integers under addition)
The End.
2 e ab2 2 4
(8)
(9) (10)
b
3
b
(11) (12) (13) (14)
e
= eeb = eb =b
A Cayley Diagram represents same info as group operation table, there is one point for each element of G, arrows represent effect of applying generator. Let’s finally talk about permutations We will see that the set of Permutations form a group under the operation of function composition. -inverse of any bijection exists ⇒ inverse permutation exists (as by definition a permutation is a bijection from a set onto itself). Cont.
Figure 1: Operation tables for Z5 Group need not be commutative (a commutative group is called Abelian after mathematician Niels Abel). Question: Can a group have more than one identity element (e)? Identity Proof. let e1 , e2 be two identity elements. What is e1 ∗ e2 ? e1 ∗ e2 = e2 (1)
Some groups have an interesting property: All elements obtained by repeatedly applying group operation to a particular element. This is called a cyclic group, element is called generator 1 = 1 mod n 1 + 1 = 2 mod n 1 + 1 + 1 = 3 mod n not every number can be the generator Cont.