User talk:Wvbailey/Function definitions

test cases for article

f is a subset of u x v

the projection of f onto u concides with all of u

each element of u corresponds to exaclty one element of v —Preceding unsigned comment added by Wvbailey (talk • contribs) 22:00, 30 September 2007 (UTC)

∀z(z ∈ f ⇒ (∃u₁ ∃v₁ (u₁∈u ⋀ v₁∈v ⋀ "z = <u₁, v₁>")))

⋀ ∀u₁ (u₁∈u ⇒ ∃z(v1∈v ⋀ "z=<u₁,v₁" ⋀ z∈f))

⋀ ∀y₁ ∀v₁ ∀v₂ (∃z₁ ∃z₂ (z₁ ∈f ⋀ z₂ ∈f ⋀ "z₁ = <u₁, v₁>" ⋀ "z₂ ⋀ <u₁, v₂>" ) ⇒ v₁ = v₂

The following is a division algorithm that, in a counter machine, produces the quotient q in register (at location) Q and residue (remainder) r at location R, i.e. [Q] = q and [R] = r. Given an n in N, it computes: n = q + r/x.

" DEFINITION 1. x is a fraction ⇔ (∃m)(∃n)(n ≠ 0 & x = <m, n>) " (Suppes Axiomatic Set Theory 1972:162)

The relation ⋍_f is defined as follows:

m₁/n₁ ⋍_f m₂/n₂ ⇔ m₁n₂ = m₂n₂

" DEFINITION 38. x is a sequence if and only if x is a function on the set ω of natural numbers." (p. 174)

" DEFINITION 40. If x is a sequence, < x₁, x₂, . . ., x_n, . . .> = x (p. 174)

"Every real number can be uniquely represented by a non-terminating decimal [i.e. made of a string of integers]" (p. 189)

"THEOREM 56. Let r be an integer ≥ 2. Every real number x is uniquiely representable with respect to the radix r as a sequence <a, d1, d2, . . ., dn, . . .> such that

"(i) a is the largest integer equal to or less than x,

"(ii) for all n, 0 ≤ dn < r and dn is an integer,

"(iii) it is not the case that there is an N fsuch that for all n > N, dn = r - 1

"(iv) the sequence whose terms cn are defined recursively by

"c0 = a

"cn+1 = cn + dn+1/rn+1

"is a Cauchy sequence which converges to x. (ibid

A "Cauchy sequence" is one that converges:

DEFINITION 44. x is a Caucy sequence of real numbers if and only if x is a sequence of real numbers and for every real number e > 0, there is a positive integer N such that for every m,n > N

|x_n - x_m| < e "(ibid p. 175)

And finally,

" DEFINITION 60. If x is a sequence of real numbers and y is the limit of x then

" lim _n→∞ x_n = y "(ibid p. 185)

"Theorem 52. A sequence of real numbers has a limit if and only if it is a Cauchy sequence." (ibid p. 185)

The

	Address	Instruction	Register	If [A]=0 goto:	If [A] ≠ 0 goto	Action	Description
	1	clr	Q	2	2	0 → Q	; CLEAR: 0 => quotient
outer_loop:	2	clr	R	3	3	0 → R	; CLEAR: 0 => residue
restore_D:	3	ldA	X	4	4	[X] → A	; LOAD_A from X: restore input X to denominator D
	4	stA	D	5	5	[A] → D	; STORE_A in D: put contents of A into denominator A
inner_loop:	5	jz	D	10	6	( [D] = 0 )*10+( [D] ≠ 0 )*6	; test D for 0: IF D=0 then go to quotient+1 step 10 else step 6
	6	jz	N	11	7	( [D] = 0 )*10+( [D] ≠ 0 )*6	; test N for 0: IF N=0 then go to done step 11 else step 7
	7	dcr	D	8	8	[D] - 1 → D	;decrement denominator D
	8	dcr	N	9	9	[N] - 1 → D	;decrement numerator N
	9	inc	R	5	5	[N] - 1 → D	;increment residue, return to inner_loop for another round
quotient+1:	10	inc	Q	2	2	[Q] - 1 → D	;increment quotient, return to outer_loop
done:	11	H					;halt