Prove (a) implies (b) implies (c) implies (d) implies (e) implies (f) implies (a)

Please just prove the first three implications {(a) implies (b) , (b) implies (c), (d) implies (e)} assuming A is c.e ( assume (a))

Problems 42-47. Prove that the following six conditions are equivalent for a language A C 10,1 (a) A is c.e. (b) There is a computable partial function f 10,10, 1 such that A dom f. (c) There is a computable partial function f : {0,1}* {0, 1)' such that A = range f. (d) A or there is a computable function f N0,1 such that A- range,f (e) A is finite or there is a computable, one-to-one function f N-[0,1 such that A- range, f (f) There is a decidable language BC 0,1 such that (Here we are using the string-pairing function z, u)-0 1xw.) Notes: . In case (d) we call f an enumerator of A, and this is the origin of the term "computably enumerable", . In case (e) we call f an enumerator of A without repetition. We sometimes write 3B for the right-hand side of the equation in (f). A string w as in (f) is called a witness that testifies that x E A. Problems 42-47. Prove that the following six conditions are equivalent for a language A C 10,1 (a) A is c.e. (b) There is a computable partial function f 10,10, 1 such that A dom f. (c) There is a computable partial function f : {0,1}* {0, 1)' such that A = range f. (d) A or there is a computable function f N0,1 such that A- range,f (e) A is finite or there is a computable, one-to-one function f N-[0,1 such that A- range, f (f) There is a decidable language BC 0,1 such that (Here we are using the string-pairing function z, u)-0 1xw.) Notes: . In case (d) we call f an enumerator of A, and this is the origin of the term "computably enumerable", . In case (e) we call f an enumerator of A without repetition. We sometimes write 3B for the right-hand side of the equation in (f). A string w as in (f) is called a witness that testifies that x E A