Mathematical Background and Basic Notations

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Colin Daniels
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1 University of Science and Technology of China (USTC) 09/19/2011

2 Outline Sets 1 Sets

3 Outline Sets 1 Sets

4 Sets Basic Notations x S membership S T subset S T proper subset S fin T finite subset N Z B the empty set natural numbers integers {true, false} S T intersection = {x x S and x T} S T union = {x x S or x T} S T difference = {x x S and x T} P(S) powerset = {T T S} [m, n] integer range = {x m x n}

5 Sets Basic Notations x S membership S T subset S T proper subset S fin T finite subset N Z B the empty set natural numbers integers {true, false} S T intersection = {x x S and x T} S T union = {x x S or x T} S T difference = {x x S and x T} P(S) powerset = {T T S} [m, n] integer range = {x m x n}

6 Generalized Unions S S(i) i I n i=m S(i) def = {x T S. x T} def = def = {S(i) i I} def = i [m,n] S(i) Here S is a set of sets. S(i) is a set whose definition depends on i. For instance, we may have S(i) = {x x > i + 3} Given i = 1, 2,...,n, we know the corresponding S(i).

7 Generalized Intersections S S(i) i I n i=m S(i) def = {x T S. x T} meaningless def = {S(i) i I} def = i [m,n] Note that is undefined. Why? S(i)

8 Some Examples A B = {A, B} Let S(i) = [i, i + 1] and I = {j 2 j [1, 3]}, then S(i) = {1, 2, 4, 5, 9, 10}. i I

9 Outline Sets 1 Sets

10 Cartesian Products The cartesian product of two sets A and B: A B def = {(x, y) x A and y B} Here (x, y) is called a pair. Projections over pairs: π 0 (x, y) = x and π 1 (x, y) = y. Generalize Cartesian products to n sets: S 0 S 1 S n 1 def = {(x 0,..., x n 1 ) x i S i, for all i [0, n 1] } We say (x 0,..., x n 1 ) is an n-tuple. Then we have π i (x 0,...,x n 1 ) = x i.

11 Sets ρ is a relation from A to B if ρ A B, or ρ P(A B). ρ is a relation on S if ρ S S. We say ρ relates x and y if (x, y) ρ. Sometimes we write it as x ρ y. ρ is an identity relation if (x, y) ρ.x = y.

12 Basic Notations the identity on S Id S {(x, x) x S} the domain of ρ dom(ρ) {x y.(x, y) ρ} the range of ρ ran(ρ) {y x.(x, y) ρ} composition of ρ and ρ ρ ρ {(x, z) y.(x, y) ρ (y, z) ρ } inverse of ρ ρ 1 {(y, x) (x, y) ρ}

13 Some Properties and Examples (ρ 3 ρ 2 ) ρ 1 = ρ 3 (ρ 2 ρ 1 ) ( ρ 1 ) 1 = ρ ρ Id S = ρ = Id T ρ (ρ 2 ρ 1 ) 1 = ρ 1 1 ρ 1 2 dom(id S ) = S = ran(id S ) Id T Id S = Id T S Id 1 S = Id S ρ = = ρ Id = = 1 dom(ρ) = ρ = < = < < = < < Id N = = = Id N = 1

14 Some Properties and Examples (ρ 3 ρ 2 ) ρ 1 = ρ 3 (ρ 2 ρ 1 ) ( ρ 1 ) 1 = ρ ρ Id S = ρ = Id T ρ (ρ 2 ρ 1 ) 1 = ρ 1 1 ρ 1 2 dom(id S ) = S = ran(id S ) Id T Id S = Id T S Id 1 S = Id S ρ = = ρ Id = = 1 dom(ρ) = ρ = < = < < = < < Id N = = = Id N = 1

15 More on Reflexivity, symmetry, and transitivity of relations. Equivalence relations on S. Total relations.

16 Outline Sets 1 Sets

17 Definition Sets A function f from A to B is a special relation from A to B such that, for all x, y and y, (x, y) f and (x, y ) f imply y = y. If f is a function, instead of writing (x, y) f, we say f x = y. and Id S are functions. If f and g are functions, then g f is a function. (g f) x = g(f x)

18 More Definitions If f is a function, f 1 is not necessarily a function. Injection, subjection, bijection. f 1 is a function if f is an injection. with types in the form of λx S. E. For instance, λx N. x + 3. Placeholder: we may use a for the bound variables: + 42 def = λx N. x + 42.

19 Variation of a function Variation of a function f{x n}: { f z if z x f{x n} = λz. n if z = x Note that x does not have to be in dom(f). dom(f{x n}) = dom(f) {x} ran(f{x n}) = (ran(f) {n f x = n and x x. f x n }) {n}

20 Function Types Sets We use A B to represent the set of all functions from A to B. is right associative. That is, A B C = A (B C). with multiple arguments: f A 1 A 2 A n A f = λx A 1 A 2 A n. E f (a 1, a 2,..., a n ) Currying of multi-argument functions: f A 1 A 2 A n A f = λx 1 A 1.λx 2 A 2....λx n A n. E f a 1 a 2... a n

21 Function Types Sets We use A B to represent the set of all functions from A to B. is right associative. That is, A B C = A (B C). with multiple arguments: f A 1 A 2 A n A f = λx A 1 A 2 A n. E f (a 1, a 2,..., a n ) Currying of multi-argument functions: f A 1 A 2 A n A f = λx 1 A 1.λx 2 A 2....λx n A n. E f a 1 a 2... a n

22 Outline Sets 1 Sets

23 Tuples as We can view a pair (x, y) as a function { x if i = 0 λi 2. y if i = 1 where 2 = {0, 1}. Similary, we can view an n-tuple (x 0,..., x n 1 ) as a function x 0 if i = 0 λi n x n 1 if i = n 1 where n = {0, 1,...,n 1}.

24 We have seen the Cartesian prodcut S 0 S n 1. We can generalize it to infinite number of sets. Let θ be an indexed family of sets θ is a function from the set of indices to a set of sets θ = {(i, S(i)) i I}, where I is the set of indices. Πθ def = {f dom(f) = dom(θ), and i dom(θ), f i θ i}

25 Example Sets Let θ = λi 2.B. Then That is, Πθ = B B. Πθ = { {(0, true),(1, true)}, {(0, true),(1, false)}, {(0, false),(1, true)}, {(0, false),(1, false)} }

26 More on We write S T for x T x T n i=m S(x) def = Πλx T. S(x) S(i) def = i [m,n] S(i) S if S is independent of x. Recall that T S is the set of all functions from T to S. So (T S) = S T

27 More on We write S T for x T x T n i=m S(x) def = Πλx T. S(x) S(i) def = i [m,n] S(i) S if S is independent of x. Recall that T S is the set of all functions from T to S. So (T S) = S T

28 Another Example A subset T of S is isomorphic with the function f T = {(x, i) x S T and i = 0 or x T and i = 1}. We know f T S 2. Then 2 S is isomorphic with P(S).

29 Another Example A subset T of S is isomorphic with the function f T = {(x, i) x S T and i = 0 or x T and i = 1}. We know f T S 2. Then 2 S is isomorphic with P(S).

30 Outline Sets 1 Sets

31 (or Disjoint Unions) A+B def = {(i, x) i = 0 and x A, or i = 1 and x B} Injection operations: ι 0 A+B A A+B ι 1 A+B B A+B The disjoint union can be generalized to n sets: S 0 + S 1 + +S n 1.

32 (or Disjoint Unions) It can also be generalized to inifinite number of sets. The disjoint union (sum) of θ is Σθ def = {(i, x) i dom(θ) and x θ i} x T n i=m S(x) def = Σλx T.S(x) S(i) def = i [m,n] S(i) S = T S if S is independent of x. x T

Outline. Sets. Relations. Functions. Products. Sums 2 / 40

Outline. Sets. Relations. Functions. Products. Sums 2 / 40 Mathematical Background Outline Sets Relations Functions Products Sums 2 / 40 Outline Sets Relations Functions Products Sums 3 / 40 Sets Basic Notations x S membership S T subset S T proper subset S fin