Cartesian Product of Functions - Formalized Mathematics

FORMALIZED

MATHEMATICS

Vol.2, No.4, September–October 1991 Universit´ e Catholique de Louvain

Cartesian Product of Functions Grzegorz Bancerek Warsaw University Bialystok

Summary. A supplement of [3] and [2], i.e. some useful and explanatory properties of the product and also the curried and uncurried functions are shown. Besides, the functions yielding functions are considered: two different products and other operation of such functions are introduced. Finally, two facts are presented: quasi-distributivity of the U f (x)

Q

≈ x X f (x) ) power of the set to other one w.r.t. the union (X x and of the poroduct w.r.t. the raising to the power Q Q quasi-distributivity ( x f (x)X ≈ ( x f (x))X ).

MML Identifier: FUNCT 6.

The articles [16], [14], [8], [17], [5], [12], [9], [11], [6], [4], [13], [15], [7], [10], [2], [1], and [3] provide the notation and terminology for this paper. Properties of Cartesian product For simplicity we follow the rules: x, y, y 1 , y2 , z, a will be arbitrary, f , g, h, h′ , f1 , f2 will denote functions, i will denote a natural number, X, Y , Z, V1 , V2 will denote sets, P will denote a permutation of X, D, D 1 , D2 , D3 will denote non-empty sets, d1 will denote an element of D1 , d2 will denote an element of D2 , and d3 will denote an element of D3 . We now state a number of propositions: Q (1) x ∈ hXi if and only if there exists y such that y ∈ X and x = hyi. Q (2) z ∈ hX, Y i if and only if there exist x, y such that x ∈ X and y ∈ Y and z = hx, yi. Q (3) a ∈ hX, Y, Zi if and only if there exist x, y, z such that x ∈ X and y ∈ Y and z ∈ Z and a = hx, y, zi. Q (4) hDi = D 1 . Q (5) hD1 , D2 i = {hd1 , d2 i}. Q (6) hD, Di = D 2 . 547

c

1991 Fondation Philippe le Hodey ISSN 0777–4028

548

grzegorz bancerek (7) (8) (9) (10)

Q

hD1 , D2 , D3 i = {hd1 , d2 , d3 i}. hD, D, Di = D 3 . Q (i 7−→ D) = D i . Q S f ⊆ ( f )dom f . Q

Curried and uncurried functions of some functions The following propositions are true: (11) If x ∈ dom f , then there exist y, z such that x = h y, zii. (12) ([: X, Y :] 7−→ z) = [: Y, X :] 7−→ z. (13) curry f = curry′ f and uncurry f = uncurry′ f . (14) If [: X, Y :] 6= ∅, then curry([: X, Y :] 7−→ z) = X 7−→ (Y 7−→ z) and curry′ ([: X, Y :] 7−→ z) = Y 7−→ (X 7−→ z). (15) uncurry(X 7−→ (Y 7−→ z)) = [: X, Y :] 7−→ z and uncurry ′ (X 7−→ (Y 7−→ z)) = [: Y, X :] 7−→ z. (16) If x ∈ dom f and g = f (x), then rng g ⊆ rng uncurry f and rng g ⊆ rng uncurry′ f . (17) dom uncurry(X 7−→ f ) = [: X, dom f :] and rng uncurry(X 7−→ f ) ⊆ rng f and dom uncurry′(X 7−→ f ) = [: dom f, X :] and rng uncurry ′ (X 7−→ f ) ⊆ rng f . (18) If X 6= ∅, then rng uncurry(X 7−→ f ) = rng f and rng uncurry ′ (X 7−→ f ) = rng f . (19) If [: X, Y :] 6= ∅ and f ∈ Z [: X, Y :] , then curry f ∈ (Z Y )X and curry′ f ∈ (Z X )Y . (20) If f ∈ (Z Y )X , then uncurry f ∈ Z [: X, Y :] and uncurry′ f ∈ Z [: Y, X :] . (21) If curry f ∈ (Z Y )X or curry′ f ∈ (Z X )Y but dom f ⊆ [: V1 , V2 :], then f ∈ Z [: X, Y :] . (22) If uncurry f ∈ Z [: X, Y :] or uncurry′ f ∈ Z [: Y, X :] but rng f ⊆ V1 →V ˙ 2 and dom f = X, then f ∈ (Z Y )X . (23) If f ∈ [: X, Y :]→Z, ˙ then curry f ∈ X →(Y ˙ →Z) ˙ and curry′ f ∈ Y →(X ˙ →Z). ˙ (24) If f ∈ X →(Y ˙ →Z), ˙ then uncurry f ∈ [: X, Y :]→Z ˙ and uncurry ′ f ∈ [: Y, X :]→Z. ˙ (25) If curry f ∈ X →(Y ˙ →Z) ˙ or curry ′ f ∈ Y →(X ˙ →Z) ˙ but dom f ⊆ [: V1 , V2 :], then f ∈ [: X, Y :]→Z. ˙ (26) If uncurry f ∈ [: X, Y :]→Z ˙ or uncurry ′ f ∈ [: Y, X :]→Z ˙ but rng f ⊆ V1 →V ˙ 2 and dom f ⊆ X, then f ∈ X →(Y ˙ →Z). ˙ 







Functions yielding functions Let X be a set. The functor Subf X is defined as follows: (Def.1) x ∈ Subf X if and only if x ∈ X and x is a function.

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Next we state four propositions: (27) Subf X ⊆ X. (28) x ∈ f −1 Subf rng f if and only if x ∈ dom f and f (x) is a function. (29) Subf ∅ = ∅ and Subf {f } = {f } and Subf {f, g} = {f, g} and Subf {f, g, h} = {f, g, h}. (30) If Y ⊆ Subf X, then Subf Y = Y . We now define three new functors. Let f be a function. The functor domκ f (κ) yielding a function is defined by: (Def.2) dom(domκ f (κ)) = f −1 Subf rng f and for every x such that x ∈ f −1 Subf rng f holds (domκ f (κ))(x) = π1 (f (x)). The functor rngκ f (κ) yields a function and is defined as follows: (Def.3) dom(rngκ f (κ)) = f −1 Subf rng f and for every x such that x ∈ f −1 Subf rng f holds (rngκ f (κ))(x) = π2 (f (x)). T The functor f is defined as follows: T T (Def.4) f = rng f . Next we state a number of propositions: (31) If x ∈ dom f and g = f (x), then x ∈ dom(dom κ f (κ)) and (domκ f (κ))(x) = dom g and x ∈ dom(rngκ f (κ)) and (rngκ f (κ))(x) = rng g. (32) domκ (κ) = and rngκ (κ) = . (33) domκ hf i(κ) = hdom f i and rngκ hf i(κ) = hrng f i. (34) domκ hf, gi(κ) = hdom f, dom gi and rngκ hf, gi(κ) = hrng f, rng gi. (35) domκ hf, g, hi(κ) = hdom f, dom g, dom hi and rng κ hf, g, hi(κ) = hrng f, rng g, rng hi. (36) domκ (X 7−→ f )(κ) = X 7−→ dom f and rngκ (X 7−→ f )(κ) = X 7−→ rng f . T (37) If f 6= , then x ∈ f if and only if for every y such that y ∈ dom f holds x ∈ f (y). T S = ∅ and = ∅. (38) S T (39) hXi = X and hXi = X. S T (40) hX, Y i = X ∪ Y and hX, Y i = X ∩ Y . S T (41) hX, Y, Zi = X ∪ Y ∪ Z and hX, Y, Zi = X ∩ Y ∩ Z. S T (42) (∅ 7−→ Y ) = ∅ and (∅ 7−→ Y ) = ∅. S T (43) If X 6= ∅, then (X 7−→ Y ) = Y and (X 7−→ Y ) = Y . Let f be a function, and let x, y be arbitrary. The functor f (x)(y) is defined by: (Def.5) f (x)(y) = (uncurry f )(hhx, yii). 













We now state several propositions: (44) If x ∈ dom f and g = f (x) and y ∈ dom g, then f (x)(y) = g(y).

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grzegorz bancerek If x ∈ dom f , then hf i(1)(x) = f (x) and hf, gi(1)(x) = f (x) and hf, g, hi(1)(x) = f (x). (46) If x ∈ dom g, then hf, gi(2)(x) = g(x) and hf, g, hi(2)(x) = g(x). (47) If x ∈ dom h, then hf, g, hi(3)(x) = h(x). (48) If x ∈ X and y ∈ dom f , then (X 7−→ f )(x)(y) = f (y). (45)

Cartesian product of functions with the same domain Q∗

Let f be a function. The functor (Def.6)

Q∗

f = curry(uncurry′ f 

f yielding a function is defined as follows:

T

[: (domκ f (κ)), dom f :]).

We now state several propositions: Q T Q Q (49) dom ∗ f = (domκ f (κ)) and rng ∗ f ⊆ (rngκ f (κ)). Q Q (50) If x ∈ dom ∗ f , then ( ∗ f )(x) is a function. Q Q (51) If x ∈ dom ∗ f and g = ( ∗ f )(x), then dom g = f −1 Subf rng f and for every y such that y ∈ dom g holds h y, xii ∈ dom uncurry f and g(y) = (uncurry f )(hhy, xii). Q (52) If x ∈ dom ∗ f , then for every g such that g ∈ rng f holds x ∈ dom g. (53) If g ∈ rng f and for every g such that g ∈ rng f holds x ∈ dom g, then Q x ∈ dom ∗ f . Q Q (54) If x ∈ dom f and g = f (x) and y ∈ dom ∗ f and h = ( ∗ f )(y), then g(y) = h(x). Q (55) QIf x ∈ dom f and f (x) is a function and y ∈ dom ∗ f , then f (x)(y) = ( ∗ f )(y)(x). Cartesian product of functions Let f be a function. The functor conditions (Def.7). Q

Q◦

f yielding a function is defined by the

Q

(Def.7) (i) dom ◦ f = (domκ f (κ)), Q (ii) Q for every g such that g ∈ (domκ f (κ)) there exists h such that ( ◦ f )(g) = h and dom h = f −1 Subf rng f and for every x such that x ∈ dom h holds h(x) = (uncurry f )(hhx, g(x)ii). The following propositions are true: Q Q (56) If g ∈ (domκ f (κ)) and x ∈ dom g, then ( ◦ f )(g)(x) = f (x)(g(x)). Q ′ = (Q◦ f )(h), (57) If x ∈ dom f and g = f (x) and h ∈ (domκ f (κ)) and h Q then h(x) ∈ dom g and h′ (x) = g(h(x)) and h′ ∈ (rngκ f (κ)). Q Q (58) rng ◦ f = (rngκ f (κ)). Q / rng f , then ◦ f is one-to-one if and only if for every g such that (59) If ∈ g ∈ rng f holds g is one-to-one. 

cartesian product of functions

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Properties of Cartesian products of functions The following propositions are true: Q∗ Q (60) = and ◦ = { } 7−→ . Q (61) dom ∗ hhi = dom h and for every x such that x ∈ dom h holds Q ( ∗ hhi)(x) = hh(x)i. Q (62) dom ∗ hf1 , fQ 2 i = dom f1 ∩dom f2 and for every x such that x ∈ dom f 1 ∩ dom f2 holds ( ∗ hf1 , f2 i)(x) = hf1 (x), f2 (x)i. Q (63) If X 6= ∅, then Q dom ∗ (X 7−→ f ) = dom f and for every x such that x ∈ dom f holds ( ∗ (X 7−→ f ))(x) = X 7−→ f (x). Q Q Q Q (64) dom ◦ hhi = hdomQ hi and rng ◦ hhi = hrng hi and for every x such that x ∈ dom h holds ( ◦ hhi)(hxi) = hh(x)i. Q Q (65) (i) dom ◦ hf1 , f2 i = hdom f1 , dom f2 i, Q◦ Q (ii) rng hf1 , f2 i = hrng f1 , rng f2 i, Q (iii) for all x, y such that x ∈ dom f 1 and y ∈ dom f2 holds ( ◦ hf1 , f2 i)(hx, yi) = hf1 (x), f2 (y)i. Q Q (66) dom ◦ (X 7−→ f ) = (dom f )X and rng Q◦ (X 7−→ f ) = (rng f )X and for every g such that g ∈ (dom f )X holds ( ◦ (X 7−→ f ))(g) = f · g. (67) If x ∈ dom f1 andQx ∈ dom f2 , then for all y1 , y2 holds hf1 , f2 i(x) = h y1 , y2 i if and only if ( ∗ hf1 , f2 i)(x) = hy1 , y2 i. (68) If x ∈ dom f1 and y ∈ dom f , then for all y1 , y2 holds [: f1 , f2 :](hhx, Q◦ 2 yii) = h y1 , y2 i if and only if ( hf1 , f2 i)(hx, yi) = hy1 , y2 i. (69) If dom f = X and domQg = X and for every x such that x ∈ X holds Q f (x) ≈ g(x), then f ≈ g. (70) If dom f = dom h and dom g = rng h and h is one-to-one and for every Q Q x such that x ∈ dom h holds f (x) ≈ g(h(x)), then f ≈ g. Q Q (71) If dom f = X, then f ≈ (f · P ). 









Function yielding powers Let us consider f , X. The functor X f yielding a function is defined by: (Def.8) dom(X f ) = dom f and for every x such that x ∈ dom f holds X f (x) = X f (x) . We (72) (73) (74) (75) (76)

now state several propositions: If ∅ ∈ / rng f , then ∅f = dom f 7−→ ∅. X = . Y hXi = hY X i. Z hX,Y i = hZ X , Z Y i. Z X7−→Y = X 7−→ Z Y . 



S

Q

(77) X disjoin f ≈ (X f ). Let us consider X, f . The functor f X yielding a function is defined by:

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grzegorz bancerek

(Def.9)

dom(f X ) = dom f and for every x such that x ∈ dom f holds f X (x) = f (x)X .

Next we state several propositions: (78) f ∅ = dom f 7−→ { }. X = . (79) (80) hY iX = hY X i. (81) hY, ZiX = hY X , Z X i. (82) (Y 7−→ Z)X = Y 7−→ Z X . Q X Q (83) (f ) ≈ ( f )X . 





References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990. Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3):537–541, 1990. Grzegorz Bancerek. K¨ onig’s theorem. Formalized Mathematics, 1(3):589–593, 1990. Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1(2):265–267, 1990. Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990. Czeslaw Byli´ nski. Basic functions and operations on functions. Formalized Mathematics, 1(1):245–254, 1990. Czeslaw Byli´ nski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990. Czeslaw Byli´ nski. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990. Czeslaw Byli´ nski. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990. Czeslaw Byli´ nski. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990. Czeslaw Byli´ nski. Partial functions. Formalized Mathematics, 1(2):357–367, 1990. Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990. Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990. Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990. Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1(3):495–500, 1990. Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990. Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97–105, 1990.

Received September 30, 1991