Problem 1. Prove that, for any two sets, A and B, the following two conditions are equivalent.

1. There isa function f:A1-+ B

2. A=φ or there is a functiong:B onto) A

Conclude that a set A is countable (as we have defined in clas) if and only if there is a function

f:AH-N.

Problem 2. Let Cn be a set of languages AS {0,1}* for eachn∈N. Prove: If each of the sets C

is countable, then sois theset∪Cn.

n=0

Problem 3. Design a Turing machine (as we defined in class) M = (Q, E, O,r,δ, s, H) that decides

the language {0°103n|n∈N}.

Problem 4. Prove that a language A C {0,1}* isc.e. if and only if there is a computable partial

function f :C {0,1}*→{0,1}* such that dom f = A.

Problem 5. Prove that, for every language A S {0, 1}*, the following conditions are equivalent.

1. A isc.e.

2. There is a computable partial function f :CN→{0,1}* such that range f= A.

3. A= φ or there is a computable function f :N→{0,1}* such that range f= A.

4. A is finite or there is a computable function f :N上{0,1}* such that range f= A.

Notes: Functions as in (3) are called enumerations of A and are the reason for the“c.e." terminol-

ogy. Functions as in (4) are called enumerations of A without repetition.

Problem 6. Prove that a language A C {0,1}* is decidable if and only if A is finite or there is a

computable function f :N→{0,1}* such that range f = A and, for every n∈N, f(n) appears

before f(n + 1) in the standard enumeration of {0, 1}*.

Problem 7. Let φ be the following statement. For every computable function f : {0,1}*→{0,1}*,

there is a computable functiong:{0,1}*→{0,1}* such that, for all x∈{0,1}*, g(f(x))=x.

1. Prove that φ is false by giving a counterexample.

2. Strengthen the hypothesis of φ just enough to obtain a statement重' that is true.

3. Prove your statement重'.

Problem 8. Prove that every infinite c.e. language A c {0, 1}* has an infinite decidable subset.

Problem 1. Prove that, for any two sets, A and B, the following two conditions are equivalent.

问题1.证明，对于任意两个集合，A和B，下列两个条件是等价的。

1. There isa function f:A1-+ B

1.有一个函数f：A1-+B

2. A=φ or there is a functiong:B onto) A

2.A=φ或在A上有一个函式：B

Conclude that a set A is countable (as we have defined in clas) if and only if there is a function

结论：集合A是可数的(正如我们在CLA中所定义的)当且仅当有一个函数

f:AH-N.

F：阿-N。

Problem 2. Let Cn be a set of languages AS {0,1}* for eachn∈N. Prove: If each of the sets C

问题2.设cn是一组语言，作为每个∈N的{0，1}*，证明：如果每个集合C

is countable, then sois theset∪Cn.

是可数的，则∪cn也是可数的。

n=0

N=0

Problem 3. Design a Turing machine (as we defined in class) M = (Q, E, O,r,δ, s, H) that decides

问题3.设计一个图灵机(如我们在类中定义的)M=(q，E，O，r，δ，S，H)

the language {0°103n|n∈N}.

语言{0°103 n=n∈N}。

Problem 4. Prove that a language A C {0,1}* isc.e. if and only if there is a computable partial

问题4.证明语言AC{0，1}*是e。当且仅当有可计算的部分

function f :C {0,1}*→{0,1}* such that dom f = A.

函数f：C{0，1}*→{0，1}*使得dom f=A。

Problem 5. Prove that, for every language A S {0, 1}*, the following conditions are equivalent.

问题5.证明，对于每种语言A S{0，1}*，下列条件是等价的。

1. A isc.e.

1.e岛。

2. There is a computable partial function f :CN→{0,1}* such that range f= A.

2.有一个可计算的部分函数f：cn→{0，1}*，其范围f=A。

3. A= φ or there is a computable function f :N→{0,1}* such that range f= A.

3.a=φ或有一个可计算函数f：n→{0，1}*，其范围f=A。

4. A is finite or there is a computable function f :N上{0,1}* such that range f= A.

4.a是有限的，或者有一个可计算函数f：n上{0，1}*，使得范围f=A。

Notes: Functions as in (3) are called enumerations of A and are the reason for the“c.e." terminol-

注：(3)中的函数称为A的枚举，是“C.E.”术语的原因-

ogy. Functions as in (4) are called enumerations of A without repetition.

奥吉。如(4)中的函数称为A的枚举，不重复。

Problem 6. Prove that a language A C {0,1}* is decidable if and only if A is finite or there is a

问题6.证明语言AC{0，1}*是可判定的当且仅当A是有限的或有

computable function f :N→{0,1}* such that range f = A and, for every n∈N, f(n) appears

可计算函数f：n→{0，1}*使范围f=A，并且，对于每n∈N，f(N)出现

before f(n + 1) in the standard enumeration of {0, 1}*.

在{0，1}*的标准枚举中f(n+1)之前。

Problem 7. Let φ be the following statement. For every computable function f : {0,1}*→{0,1}*,

问题7.让φ成为下面的语句。对于每个可计算函数f：{0，1}*→{0，1}*，

there is a computable functiong:{0,1}*→{0,1}* such that, for all x∈{0,1}*, g(f(x))=x.

有一个可计算的函数：{0，1}*→{0，1}*，使得对于所有x∈{0，1}*，g(f(X))=x。

1. Prove that φ is false by giving a counterexample.

1.给出一个反例，证明φ是假的。

2. Strengthen the hypothesis of φ just enough to obtain a statement重' that is true.

2.加强φ的假设，以获得重‘是正确的陈述。

3. Prove your statement重'.

3.证明你的陈述是重‘。

Problem 8. Prove that every infinite c.e. language A c {0, 1}* has an infinite decidable subset.

问题8.证明每一个无限的C.E.语言Ac{0，1}*有一个无穷可判定子集。