The Cantor Game: Winning Strategies and Determinacy

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by  Magnus D. LaDue

THE CANTOR GAME: WINNING
STRATEGIES AND DETERMINACY
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0. Abstract
In [1] Grossman and Turett define the Cantor game. In [2] Matt Baker
proves several results about the Cantor game and poses three challenging
questions about it:
Do there exist uncountable subsets of [0,1] for which:
(1) Alice does not have a winning strategy;
(2) Bob has a winning strategy;
(3) neither Alice nor Bob has a winning strategy?
In this paper we show that the answers to these questions depend upon
whichaxiomsofsettheoryareassumed. Specifically,ifweassumetheAxiom
ofDeterminacyinadditiontotheZermelo-Fraenkelaxioms, thentheanswer
to all three questions is “no.” If instead we assume the Zermelo-Fraenkel
axioms together with the Axiom of Choice, then the answer to questions 1
and 3 is “yes,” and the answer to question 2 is likely to be “no.”
Author’s Note: This paper was my entry in the 2017 Regeneron Science
Talent Search. It earned a Top 300 Scholar Award as well as Research
Report and Student Initiative badges.
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Figure 1. A play of the Cantor game. Player A wins if a ∈ S.
1. Introduction
In their paper [1] Grossman and Turett define the Cantor game. In [2]
Matt Baker proves several results about the Cantor game and poses several
challenging questions about it. The Cantor game is an infinite game played
on the real line by two players, A (Alice) and B (Bob). Let a and b
0 0
be fixed real numbers such that a < b , and in the interval [a ,b ] fix a
0 0 0 0
subset S, which we call the target set. (In the original version of the Cantor
game, a and b are chosen to be 0 and 1, respectively. In this case we take
0 0
them to be arbitrary real numbers.) Initially, player A chooses a , and then
0
player B chooses b . On the nth turn, for n ≥ 1, player A chooses a with
0 n
the property that a < a < b , and then player B chooses b such
n−1 n n−1 n
that a < b < b . Since the sequence (a ) is strictly increasing and
n n n−1 n n≥0
bounded above by b , lim a = a exists, and since (b ) is strictly
0 n→∞ n n n≥0
decreasing and bounded below by a , lim b = b exists. Since a < b
0 n→∞ n n n
for all n, we also have a ≤ b. In the end, if a ∈ S, player A wins; otherwise,
if a ∈/ S, player B wins. Figure 1 illustrates a play of the Cantor game.
Baker uses the Cantor game to prove that the closed interval [0,1] is
uncountable [2, p.377] and that every perfect set is uncountable [2, p.378].
He also shows that player B has a winning strategy when the target set S is
countable [2, p.377]. After proving these results he poses three challenging
questions [2, p.379]:
Do there exist uncountable subsets of [0,1] for which:
(1) Alice does not have a winning strategy;
(2) Bob has a winning strategy;
(3) neither Alice nor Bob has a winning strategy?
We will show that the answers to these questions depend upon which
axioms of set theory we assume. First we assume only the Zermelo-Fraenkel
axioms of set theory (ZF) and prove some results about winning strategies
for the Cantor game. Then assuming the Axiom of Determinacy in addition
to ZF (ZF + AD), we shall prove that the answer to all three questions is
“no.” Finally, assuming the Axiom of Choice in addition to ZF (ZFC), we
will show that the answer to questions 1 and 3 is “yes,” and we will give
evidence that the answer to question 2 is likely to be “no.”
2
2. Strategic definitions
ThomasJechbeginshismonographonaxiomaticsettheorybydiscussing
the eight Zermelo-Fraenkel (ZF) axioms. He then develops some conse-
quencesofthoseaxiomsandshowshowtherealnumberscanbeconstructed
without additional assumptions [4]. This allows us to see which standard
resultsofelementaryrealanalysisfollowfromtheZFaxioms. Inthissection
and the following two sections we assume only the eight ZF axioms. Once
we have proved several fascinating results about the Cantor game, we shall
assume additional axioms and provide answers to our three questions.
Because our answers will require precise definitions and properties of
strategies and generalized Cantor sets, we begin by defining these funda-
mental concepts. Our first definition was inspired by Oxtoby’s work on the
related Banach-Mazur game in [3, p.27]. In the following definitions we con-
sider the Cantor game on [a ,b ], where a and b are fixed real numbers
0 0 0 0
satisfying a < b , with an uncountable target set S.
0 0
Definition 1. A strategy for player A is an infinite sequence of real-valued
functions (f ) . The function f is required to have the domain {(a ,b )},
n n≥0 0 0 0
and its value must satisfy a < f (a ,b ) < b . For each n ≥ 2 the function
0 0 0 0 0
f has domain {(a ,b ,a ,b ,...,a ,b ) | a < a < ··· < a <
n−1 0 0 1 1 n−1 n−1 0 1 n−1
b < b < ··· < b }. If a = f (a ,b ,...,a ,b ), then we
n−1 n−2 0 n n−1 0 0 n−1 n−1
require that a < a < b .
n−1 n n−1
Likewise, a strategy for player B is a sequence of real-valued functions
(g ) . The function g is required to have the domain {(a ,b ,a ) | a <
n n≥0 0 0 0 1 0
a < b }, and its value must satisfy a < g (a ,b ,a ) < b . For each
1 0 1 0 0 0 1 0
n ≥ 2 the function g has domain {(a ,b ,a ,b ,...,a ,b ,a ) |
n−1 0 0 1 1 n−1 n−1 n
a < a < ··· < a < b < b < ··· < b }. Furthermore, if
0 1 n n−1 n−2 0
b = g (a ,b ,...,a ,b ,a ), then we require that a < b < b .
n n−1 0 0 n−1 n−1 n n n n−1
Definition 2. A play of the game is an ordered pair of sequences ((a ) ,
n n≥0
(b ) ), where a < a < b < b for each n ≥ 1. The play is
n n≥0 n−1 n n n−1
consistent with a given strategy for player A (f ) if and only if a =
n n≥0 n
f (a ,b ,a ,b ,...,a ,b ) for all n ≥ 1. Similarly, the play is con-
n−1 0 0 1 1 n−1 n−1
sistent with the strategy for player B (g ) if and only if b = g (a ,b ,
n n≥0 n n−1 0 0
...,a ,b ,a ) for all n ≥ 1.
n−1 n−1 n
Definition 3. The limit set of a strategy (f ) for player A is defined as
n n≥0
follows:
L((f )) = { lim a | ((a ),(b )) is a play of the game consistent with (f )}.
n n n n n
n→∞
Likewise, the limitsetofastrategy(g ) forplayerBisdefinedasfollows:
n n≥0
L((g )) = { lim a | ((a ),(b )) is a play of the game consistent with (g )}.
n n n n n
n→∞
We define a strategy (f ) for player A to be a winning strategy if and
n n≥0
only if L((f )) ⊂ S. A strategy (g ) for player B is said to be a winning
n n n≥0
strategy if and only if L((g )) ⊂ [a ,b ]−S.
n 0 0
3
3. Generalized Cantor sets
We define the concept of a generalized Cantor set as follows.
Definition 4. Let I = [c,d], where c < d, and suppose that (e ) is a
n n≥1
strictly decreasing sequence of positive real numbers that converges to 0. For
each n ≥ 1 and each of the 2n finite sequences of 0s and 1s, (i ,i ,...,i ),
1 2 n
let I = [c ,d ] be a closed interval. Suppose that the
i1,i2,...,in i1,i2,...,in i1,i2,...,in
following properties hold:
(1) 0 < d −c < e for all n ≥ 1 and all i ,...,i ∈ {0,1};
i1,...,in i1,...,in n 1 n
(2) I ,I ⊂ I and I ⊂ I for all n ≥ 2 and all i ,...,i ∈
0 1 i1,...,in i1,...,in−1 1 n
{0,1};
(3) I ∩ I = ∅ and I ∩ I = ∅ for all n ≥ 2 and all
0 1 i1,...,in−1,0 i1,...,in−1,1
i ,...,i ∈ {0,1}.
1 n−1
For each n ≥ 1 let
(cid:91)
C = I ,
n i1,i2,...,in
i1,...,in∈{0,1}
and define
(cid:92)
C = C .
n
n≥1
Then C is called the generalized Cantor set generated by the collection of
intervals
C = {I | n ≥ 1 and i ,...,i ∈ {0,1}}.
i1,...,in 1 n
GeneralizedCantorsetshavemanyofthefamiliarpropertiesoftheCantor
middle-thirdsset. BeforestatingourmainresultsongeneralizedCantorsets,
we first prove a lemma.
Lemma 1. Let C be the generalized Cantor set generated by the collection
of intervals C in Definition 4. Then for any n ≥ 1, if I ∩I (cid:54)= ∅,
i1,...,in j1,...,jn
then i = j ,...,i = j .
1 1 n n
Proof. Whenn = 1,ifi (cid:54)= j ,thenI ∩I = ∅,andsotheresultholds. Now
1 1 i1 j1
assume that the result holds for n−1. Suppose that x ∈ I ∩I .
i1,...,in j1,...,jn
By (2) of Definition 4 we see that x ∈ I ∩I . Now, by the
i1,...,in−1 j1,...,jn−1
induction hypothesis, i = j ,...,i = j . Hence x ∈ I ∩
1 1 n−1 n−1 i1,...,in−1,in
I . By (3) of Definition 4, these intervals are disjoint if i (cid:54)= j . It
i1,...,in−1,jn n n
follows that i = j , and so the result is true by induction. (cid:3)
n n
Theorem 1. Let C be the generalized Cantor set generated by the collection
of intervals C in Definition 4. Then the following properties hold:
(1) If (i ) is any sequence in {0,1}, then lim c =
n n≥1 n→∞ i1,...,in
lim d ;
n→∞ i1,...,in
(2) x ∈ C if and only if there is a unique sequence (i ) in {0,1} such
n n≥1
(cid:84)
that {x} = I . Furthermore, since c ≤ x ≤ d
n≥1 i1,...,in i1,...,in i1,...,in
for all n and the two sequences have equal limits, we must have
lim c = x = lim d ;
n→∞ i1,...,in n→∞ i1,...,in
4
(3) C is a perfect set.
Proof. (1) Let (i ) be any sequence in {0,1}. By (2) of Definition 4,
n n≥1
I ⊂ I , so that c ≤ c . Hence (c )
i1,...,in i1,...,in−1 i1,...,in−1 i1,...,in i1,...,in n≥1
is a monotonically increasing sequence bounded above by d, which
implies that lim c exists. A similar argument shows that
n→∞ i1,...,in
lim d exists. Since lim e = 0 and 0 < d −
n→∞ i1,...,in n→∞ n i1,...,in
c < e by (1) of Definition 4, we have lim (d −
i1,...,in n n→∞ i1,...,in
c ) = 0, and so lim c = lim d .
i1,...,in n→∞ i1,...,in n→∞ i1,...,in
(2) Suppose that x ∈ C; then x ∈ C for all n ≥ 1. We proceed by
n
induction on n. When n = 1, we know that x ∈ C = I ∪I , and we
1 0 1
choosei ∈ {0,1}suchthatx ∈ I . Nowassumethati ,i ,...,i
1 i1 1 2 n−1
have been chosen such that x ∈ I ,x ∈ I ,...,x ∈ I .
i1 i1,i2 i1,...,in−1
Since x ∈ C , there exist j ,j ,...,j ,i ∈ {0,1} such that x ∈
n 1 2 n−1 n
I . By (2) of Definition 4, x ∈ I . By Lemma 1
j1,...,jn−1,in j1,...,jn−1
this implies that i = j ,...,i = j . Thus x ∈ I . By
1 1 n−1 n−1 i1,...,in
induction, it follows that we have defined a sequence (i ) such
n n≥1
(cid:84)
that x ∈ I .
n≥1 i1,...,in
(cid:84)
Next, assume that y ∈ I . Then x,y ∈ [c ,d ]
n≥1 i1,...,in i1,...,in i1,...,in
for n ≥ 1, and so |y −x| ≤ d −c < e for all n. Since
i1,...,in i1,...,in n
(e ) converges to 0, it follows that |y − x| = 0, and so y = x.
n n≥1
(cid:84)
Thus there exists a sequence (i ) such that {x} = I .
n n≥1 n≥1 i1,...,in
Now assume that there is another sequence (j ) having the same
n n≥1
property, and let n ≥ 1 be given. Then x ∈ I ∩I , and so
i1,...,in j1,...,jn
by Lemma 1, we have i = j . Since n ≥ 1 was arbitrary, it follows
n n
that the two sequences are identical. This shows that the sequence
(i ) is uniquely determined.
n n≥1
Conversely, assume that there is a sequence (i ) such that
n n≥1
(cid:84)
{x} = I . Since x ∈ I ⊂ C for each n ≥ 1, it
n≥1 i1,...,in i1,...,in n
follows that x ∈ C, thereby completing the proof.
(3) Since I is closed, each C is a finite union of closed sets, so
i1,...,in n
that each C is closed. So C is an intersection of closed sets, and it
n
follows that C is closed. Now assume that x ∈ C. Then there is a
(cid:84)
sequence (i ) such that {x} = I . Let (cid:15) > 0 be given.
n n≥1 n≥1 i1,...,in
Then there exists N such that e < (cid:15). Define a sequence (j ) as
N n n≥1
follows:
(cid:40)
1−i if n = N +1;
n
j =
n
i if n (cid:54)= N +1.
n
(cid:84)
Let y be such that {y} = I . Then y ∈ C, and by part
n≥1 j1,...,jn
(2), y (cid:54)= x. Observe that x,y ∈ I ; hence |x−y| ≤ d −
i1,...,iN i1,...,iN
c < e < (cid:15), so that y ∈ (x−(cid:15),x+(cid:15)). This shows that x is a
i1,...,iN N
limit point of C. Since x was arbitrary, it follows that C is a perfect
set.
(cid:3)
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4. The Cantor Game and the Zermelo-Fraenkel Axioms
Using our precise definitions of strategies for players A and B and those
properties of generalized Cantor sets given by Theorem 1, we can now prove
some remarkably powerful results about limit sets for both players. Once
again we note that we are assuming only the axioms of ZF.
First we will show that the limit set of any strategy for player A contains
a generalized Cantor set. Then we will use this result to characterize those
target sets for which player A has a winning strategy. Next we will show
that the limit set of any strategy for player B also contains a generalized
Cantor set, and we will use this result to provide necessary conditions for
player B to have a winning strategy. Finally, we will introduce the Perfect
Set Property and show that it has a close relationship to the Cantor game.
4.1. Results for player A.
Theorem 2. In the Cantor game on [a ,b ] with an uncountable target set
0 0
S, the limit set of any strategy for player A contains a generalized Cantor
set.
Proof. Suppose that the uncountable target set S for the Cantor game on
[a ,b ] is given, and let (f ) be a strategy for player A. Assume that we
0 0 n n≥0
have constructed a fixed enumeration Q of the rational numbers in [a ,b ].
0 0 0
Lete = b0−a0 forn ≥ 1, anddefinea = f (a ,b ), c = a , d = b , u = c+d,
n 2n 1 0 0 0 1 0 2
and I = [c,d]. This defines the interval and the bounding sequence that we
will use for the construction of a generalized Cantor set.
Let d be the first element of Q (the one of least index) such that c <
1 0
d < u, and let
1
c = f (a ,b ,c,d ).
1 1 0 0 1
Define d to be the first element of Q such that c < d < c , and let
0 0 0 1
c = f (a ,b ,c,d ).
0 1 0 0 0
Also define
I = [c ,d ];
0 0 0
I = [c ,d ];
1 1 1
c +d
0 0
u = ;
0
2
and
c +d
1 1
u = .
1
2
Then we have
c < c < d < c < d < u < d.
0 0 1 1
Hence
I ,I ⊂ [c,u] ⊂ I
0 1
and
I ∩I = ∅.
0 1
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By the preceding inequality we have
d−c b −a b −a
0 1 0 0
0 < d −c < u−c = = < = e
i1 i1 2 2 2 1
for i ∈ {0,1}. Observe that the left endpoints of I and I are defined by
1 0 1
using player A’s strategy based upon two rational numbers that might be
chosen by player B.
Next we define
d = first element of Q such that c < d < u ;
1,1 0 1 1,1 1
c = f (a ,b ,c,d ,c ,d );
1,1 2 0 0 1 1 1,1
d = first element of Q such that c < d < c ;
1,0 0 1 1,0 1,1
c = f (a ,b ,c,d ,c ,d );
1,0 2 0 0 1 1 1,0
d = first element of Q such that c < d < u ;
0,1 0 0 0,1 0
c = f (a ,b ,c,d ,c ,d );
0,1 2 0 0 0 0 0,1
d = first element of Q such that c < d < c ;
0,0 0 0 0,0 0,1
and
c = f (a ,b ,c,d ,c ,d ).
0,0 2 0 0 0 0 0,0
Let
I = [c ,d ]
i1,i2 i1,i2 i1,i2
and
c +d
u = i1,i2 i1,i2
i1,i2 2
for i ,i ∈ {0,1}. Observe once again that the left endpoint of each interval
1 2
I is defined by using player A’s strategy based upon a rational number
i1,i2
that might be chosen by player B.
Now assume that for each k with 2 ≤ k ≤ n, the collections of real
numbers {c | i ,...,i ∈ {0,1}}, {d | i ,...,i ∈ {0,1}}, and
i1,...,ik 1 k i1,...,ik 1 k
{u | i ,...,i ∈ {0,1}} and the collection of closed intervals {I |
i1,...,ik 1 k i1,...,ik
i ,...,i ∈ {0,1}} have been defined and satisfy the following properties for
1 k
all choices of i ,i ,...,i ∈ [0,1]:
1 2 k
(1) d = first element of Q belonging to (c ,u );
i1,...,ik−1,1 0 i1,...,ik−1 i1,...,ik−1
(2) c = f (a ,b ,c,d ,c ,...,d ,c ,d );
i1,...,ik−1,1 k 0 0 i1 i1 i1,...,ik−1 i1,...,ik−1 i1,...,ik−1,1
(3) d = first element of Q belonging to (c ,c );
i1,...,ik−1,0 0 i1,...,ik−1 i1,...,ik−1,1
(4) c = f (a ,b ,c,d ,c ,...,d ,c ,d );
i1,...,ik−1,0 k 0 0 i1 i1 i1,...,ik−1 i1,...,ik−1 i1,...,ik−1,0
(5) I = [c ,d ]; and
i1,...,ik i1,...,ik i1,...,ik
(6) u = ci1,...,ik+di1,...,ik.
i1,...,ik 2
Then define collections of real numbers {c | i ,...,i ∈ {0,1}},
i1,...,in+1 1 n+1
{d | i ,...,i ∈ {0,1}}, and {u | i ,...,i ∈ {0,1}}, as
i1,...,in+1 1 n+1 i1,...,in+1 1 n+1
well as a collection of closed intervals {I | i ,...,i ∈ {0,1}} with
i1,...,in+1 1 n+1
the following properties for each choice of i ,i ,...,i ∈ [0,1]:
1 2 n+1
(1) d = first element of Q belonging to (c ,u );
i1,...,in,1 0 i1,...,in i1,...,in
(2) c = f (a ,b ,c,d ,c ,...,d ,c ,d );
i1,...,in,1 n+1 0 0 i1 i1 i1,...,in i1,...,in i1,...,in,1
(3) d = first element of Q belonging to (c ,c );
i1,...,in,0 0 i1,...,in i1,...,in,1
7
(4) c = f (a ,b ,c,d ,c ,...,d ,c ,d );
i1,...,in,0 n+1 0 0 i1 i1 i1,...,in i1,...,in i1,...,in,0
(5) I = [c ,d ]; and
i1,...,in+1 i1,...,in+1 i1,...,in+1
(6) u = ci1,...,in+1+di1,...,in+1.
i1,...,in+1 2
Byinductionthefirstsetofsixpropertiesholdsforthecollectionsofnumbers
andintervalsdefinedforeveryk ≥ 2andeverychoiceofi ,i ,...,i ∈ [0,1].
1 2 k
Now fix n ≥ 2, and let k = n. Applying properties (1) - (4) together
with the inequality for f in Definition 2, we see that for all choices of
n
i ,i ,...,i ∈ [0,1],
1 2 n−1
c < c < d < c
i1,...,in−1 i1,...,in−1,0 i1,...,in−1,0 i1,...,in−1,1
and
c < d < u < d .
i1,...,in−1,1 i1,...,in−1,1 i1,...,in−1 i1,...,in−1
Hence
I ,I ⊂ [c ,u ] ⊂ I
i1,...,in−1,0 i1,...,in−1,1 i1,...,in−1 i1,...,in−1 i1,...,in−1
and
I ∩I = ∅.
i1,...,in−1,0 i1,...,in−1,1
Combining these relations with those of the construction for n = 1, we see
that conditions (2) and (3) of Definition 4 are satisfied.
To see that condition (1) of that definition also holds, we proceed by
induction. Recall that we have defined e = b0−a0 for n ≥ 1, and we have
n 2n
already shown that 0 < d − c < e for i ∈ {0,1}. Now assume that
i1 i1 1 1
0 < d −c < e for all choices of i ,...,i ∈ {0,1}, and
i1,...,in−1 i1,...,in−1 n−1 1 n−1
let i ,...,i ∈ {0,1} be given. Then d −c > 0 and
1 n i1,...,in i1,...,in
d −c e
d −c < u −c = i1,...,in−1 i1,...,in−1 < n−1 = e .
i1,...,in i1,...,in i1,...,in−1 i1,...,in−1 2 2 n
Hence condition (1) of Definition 4 holds.
Now let C be the generalized Cantor set generated by C = {I |
i1,...,in
n ≥ 1 and i ,...,i ∈ {0,1}}. We claim that C ⊂ L((f ) ). To prove
1 n n n≥0
this, let x ∈ C be given. By Theorem 1 there is a unique sequence (i )
n n≥1
(cid:84)
such that {x} = I . We define a play of the Cantor game as
n≥1 i1,...,in
follows. First note that a and b are already given. Now let a = c,
0 0 1
b = d , and define a = c and b = d for all n ≥ 2. Then
1 i1 n i1,...,in−1 n i1,...,in
a = f (a ,b ), a = c = f (a ,b ,c,d ) = f (a ,b ,a ,b ), and for n ≥ 3,
1 0 0 0 2 i1 1 0 0 i1 1 0 0 1 1
a = c = f (a ,b ,c,d ,c ,...,d ,c ,d ) =
n i1,...,in−1 n−1 0 0 i1 i1 i1,...,in−2 i1,...,in−2 i1,...,in−1
f (a ,b ,a ,b ,a ,...,b ,a ,b ) = f (a ,b ,a ,b ,...,a ,
n−1 0 0 1 1 2 n−2 n−1 n−1 n−1 0 0 1 1 n−1
b ). Thus the play of the game ((a ) ,(b ) ) is consistent with
n−1 n n≥0 n n≥0
the given strategy (f ) for player A. Now observe that lim a =
n n≥0 n→∞ n
lim c = x, so that x ∈ L((f ) ). It follows that the general-
n→∞ i1,...,in−1 n n≥0
ized Cantor set we have defined is a subset of player A’s limit set. (cid:3)
As a consequence of Theorem 2 we obtain a characterization of those
target sets S for which player A has a winning strategy.
8
Theorem 3. In the Cantor game on [a ,b ] with an uncountable target set
0 0
S, the following three statements are equivalent:
(1) Player A has a winning strategy;
(2) S contains a generalized Cantor set;
(3) S contains a perfect set.
Proof. (1) ⇒ (2) If player A has a winning strategy, then S contains the
limitsetofthatstrategy. ByTheorem2,thislimitsetcontainsageneralized
Cantor set. Thus S contains a generalized Cantor set.
(2) ⇒ (3) This is true according to Theorem 1 since every generalized
Cantor set is a perfect set.
(3) ⇒ (1) This was proved in [2, p.379]. (cid:3)
4.2. Results for player B. We now prove that similar results hold for
player B’s strategy, although the implications of these results will be less
far-reachingthantheresultsforplayerA.Theproofofournextresultisvery
similar to the proof of Theorem 2 for player A. The main difference is that
the current argument’s construction proceeds upward from the midpoint of
each subinterval, while the construction in Theorem 2 proceeds downward.
Theorem 4. In the Cantor game on [a ,b ] with an uncountable target set
0 0
S, the limit set of any strategy for player B contains a generalized Cantor
set.
Proof. Suppose that the uncountable target set S for the Cantor game on
[a ,b ] is given, and let (g ) be a strategy for player B. Assume that we
0 0 n n≥0
have constructed a fixed enumeration Q of the rational numbers in [a ,b ].
0 0 0
Let e = b0−a0 for n ≥ 1, and define c = a , d = b , u = c+d, and I = [c,d].
n 2n 0 0 2
This defines the interval and the bounding sequence that we will use for the
construction of a generalized Cantor set.
Let c be the first element of Q (the one of least index) such that u <
0 0
c < d, and let
0
d = g (a ,b ,c ).
0 0 0 0 0
Define c to be the first element of Q such that d < c < d, and let
1 0 0 1
d = g (a ,b ,c ).
1 0 0 0 1
Also define
I = [c ,d ];
0 0 0
I = [c ,d ];
1 1 1
c +d
0 0
u = ;
0
2
and
c +d
1 1
u = .
1
2
Then we have
c < u < c < d < c < d < d.
0 0 1 1