König's Theorem - Formalized Mathematics

FORMALIZED MATHEMATICS Vol.1, No.3, May–August 1990 Universit´ e Catholique de Louvain

K¨ onig’s Theorem Grzegorz Bancerek Warsaw University Bialystok

Summary. In the article the sum and product of any number of cardinals are introduced and their relationships to addition, multiplication and to other concepts are shown. Then the K¨ onig’s theorem is proved. The theorem that the cardinal of union of increasing family of sets of power less than some cardinal m is not greater than m, is given too.

MML Identifier: CARD 3.

The papers [12], [6], [7], [3], [14], [13], [4], [2], [11], [9], [8], [10], [1], and [5] provide the terminology and notation for this paper. For simplicity we adopt the following rules: A, B are ordinal numbers, K, M , N are cardinal numbers, x, y, z are arbitrary, X, Y , Z, Z1 , Z2 are sets, n is a natural number, and f , g are functions. A function is said to be a function yielding cardinal numbers if: for every x such that x ∈ dom it holds it(x) is a cardinal number. Next we state a proposition (1) f is a function yielding cardinal numbers if and only if for every x such that x ∈ dom f holds f (x) is a cardinal number. In the sequel f f denotes a function yielding cardinal numbers. Let us consider f f , X. Then f f X is a function yielding cardinal numbers. Let us consider f f , x. Then f f (x) is a set. Let us consider X, K. Then X 7−→ K is a function yielding cardinal numbers. The following propositions are true: (2) f f X is a function yielding cardinal numbers and X 7−→ K is a function yielding cardinal numbers. is a function yielding cardinal numbers. (3) The scheme CF Lambda concerns a set A and a unary functor F yielding a cardinal number and states that: 589

c

1990 Fondation Philippe le Hodey ISSN 0777–4028

590

Grzegorz Bancerek there exists f f such that dom f f = A and for every x such that x ∈ A holds f f (x) = F(x) for all values of the parameters. We now define four new functors. Let us consider f . The functor f yields a function yielding cardinal numbers and is defined as follows: dom f = dom f and for every x such that x ∈ dom f holds f (x) = [f (x)]. The functor disjoin f yielding a function, is defined as follows: dom(disjoin f ) = dom f and for every x such that x ∈ dom f holds (disjoin f )(x) = [: [f (x)], {x} :] . S The functor f yields a set and is defined by: S S f = (rng f ). Q The functor f yielding a set, is defined by: Q x ∈ f if and only if there exists g such that x = g and dom g = dom f and for every x such that x ∈ dom f holds g(x) ∈ [f (x)]. We now state a number of propositions: (4)

f f = f if and only if dom f f = dom f and for every x such that x ∈ dom f holds f f (x) = [f (x)] . (5) g = disjoin f if and only if dom g = dom f and for every x such that x ∈ dom f holds g(x) = [: [f (x)], {x} :]. S S (6) f = (rng f ). Q (7) X = f if and only if for every x holds x ∈ X if and only if there exists g such that x = g and dom g = dom f and for every x such that x ∈ dom f holds g(x) ∈ [f (x)]. (8)

ff = ff.

(9)

= .

X 7−→ Y = X 7−→ Y . disjoin = . disjoin({x} 7−→ X) = {x} 7−→ [: X, {x} :]. If x ∈ dom f and y ∈ dom f and x 6= y, then [disjoin f (x)] ∩ [disjoin f (y)] = ∅ . S (14) = ∅. S (15) (X 7−→ Y ) ⊆ Y . S (16) If X 6= ∅, then (X 7−→ Y ) = Y . S (17) ({x} 7−→ Y ) = Y . Q (18) g ∈ f if and only if dom g = dom f and for every x such that x ∈ dom f holds g(x) ∈ [f (x)]. Q = { }. (19) Q X (20) Y = (X 7−→ Y ). Let us consider x, X. The functor πx X yields a set and is defined by: y ∈ πx X if and only if there exists f such that f ∈ X and y = f (x). (10) (11) (12) (13)

¨ nig’s Theorem Ko

591

Next we state a number of propositions: (21) Y = πx X if and only if for every y holds y ∈ Y if and only if there exists f such that f ∈ X and y = f (x). Q Q (22) If x ∈ dom f and f 6= ∅, then πx ( f ) = f (x). (23) If f ∈ X, then f (x) ∈ πx X. (24) πx ∅ = ∅. (25) πx {g} = {g(x)}. (26) πx {f, g} = {f (x), g(x)}. (27) πx (X ∪ Y ) = πx X ∪ πx Y . (28) πx (X ∩ Y ) ⊆ πx X ∩ πx Y . (29) πx X \ πx Y ⊆ πx (X \ Y ). . π Y ⊆ π (X− . Y ). (30) πx X− x x πx X ≤ X . S If x ∈ (disjoin f ), then there exist y, z such that x = h y, zii. S x ∈ (disjoin f ) if and only if x2 ∈ dom f and x1 ∈ [f (x2 )] and x = h x 1 , x2 i . (34) If f ≤ g, then disjoin f ≤ disjoin g. S S (35) If f ≤ g, then f ⊆ g. S (36) (disjoin(Y 7−→ X)) = [: X, Y :]. Q (37) f = ∅ if and only if ∅ ∈ rng f . (38) If dom f =Qdom gQand for every x such that x ∈ dom f holds [f (x)] ⊆ [g(x)], then f ⊆ g. In the sequel F , G will denote functions yielding cardinal numbers. The following two propositions are true: (31) (32) (33)

(39)

For every x such that x ∈ dom F holds F (x) = F (x).

(40) For every x such that x ∈ dom F holds [disjoin F (x)] = F (x). P We now define two new functors. Let us consider F . The functor F yields a cardinal number and is defined as follows: P S F = (disjoin F ). Q The functor F yielding a cardinal number, is defined as follows: Q Q F = F. The following propositions are true: (41)

P

Q

F =

S

(disjoin F ).

Q

F = F. If dom F P = dom G and for every x such that x ∈ dom F holds F (x) ⊆ P G(x), then F ≤ G. Q (44) ∅ ∈ rng F if and only if F = 0. (45) If dom F Q = dom Q G and for every x such that x ∈ dom F holds F (x) ⊆ G(x), then F ≤ G.

(42) (43)

592

Grzegorz Bancerek (46) (47) (48) (49) (50) (51) (52) (53) (54)

P

P

If F ≤ G, then F ≤ G. Q Q If F ≤ G and 0 ∈ / rng G, then F ≤ G. P (∅ 7−→ K) = 0. Q (∅ 7−→ K) = 1. P ({x} 7−→ K) = K. Q ({x} 7−→ K) = K. P (M 7−→ N ) = M · N . Q (N 7−→ M ) = M N . S S

f ≤

P

P

f.

F ≤ F. If dom F P = dom G and for every x such that x ∈ dom F holds F (x) ∈ Q G(x), then F < G. Now we present three schemes. The scheme FinRegularity deals with a set A, and a binary predicate P, and states that: there exists x such that x ∈ A and for every y such that y ∈ A and y 6= x holds not P[y, x] provided the following conditions are fulfilled: • A is finite and A 6= ∅, • for all x, y such that P[x, y] and P[y, x] holds x = y, • for all x, y, z such that P[x, y] and P[y, z] holds P[x, z]. The scheme MaxFinSetElem concerns a set A, and a binary predicate P, and states that: there exists x such that x ∈ A and for every y such that y ∈ A holds P[x, y] provided the following requirements are fulfilled: • A is finite and A 6= ∅, • for all x, y holds P[x, y] or P[y, x], • for all x, y, z such that P[x, y] and P[y, z] holds P[x, z]. The scheme FuncSeparation deals with a set A, a unary functor F yielding a set, and a binary predicate P, and states that: there exists f such that dom f = A and for every x such that x ∈ A for every y holds y ∈ [f (x)] if and only if y ∈ F(x) and P[x, y] for all values of the parameters. We now state several propositions: (57) Rord(n) is finite. (55) (56)

(58)

If X is finite, then X < ω .

(59)

If A < B , then A ∈ B.

(60) (61)

If A < M , then A ∈ M . Suppose for all Z1 , Z2 such that Z1 ∈ X and Z2 ∈ X holdsSZ1 ⊆ Z2 or S Z2 ⊆ Z1 . Then there exists Y such that Y ⊆ X and Y = X and for every Z such that Z ⊆ Y and Z 6= ∅ there exists Z 1 such that Z1 ∈ Z and for every Z2 such that Z2 ∈ Z holds Z1 ⊆ Z2 .

¨ nig’s Theorem Ko (62)

593

If for every Z such that Z ∈ X holds Z < M and for all Z 1 , Z2 such S that Z1 ∈ X and Z2 ∈ X holds Z1 ⊆ Z2 or Z2 ⊆ Z1 , then X ≤ M .

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