Recall that we have used the string-pairing function (:,:): {0,1}* > {0,1}* {0,1}* defined by (x, y) = 011|1xy for all x, y {0,1}* Given a language B C {0,1}*, define the language 3B C {0,1}* by 3B = {x {0,1}* |(Ew e {0,1}*)(x, W) E B}. Problems 50 and 51 characterize computable enumerability in terms of unbounded existential search. Problem 50. Let B C {0,1}*. Prove: If B is c.e., then 3B is c.e. Problem 51. Prove: For every c.e. language A C{0,1}*, there is a decidable language B C {0,1}* such that A= B. Recall that the join of two languages A, B C {0,1}* is AU B = {x0 | X A} U{y1|y E B}. Problem 52. Let A, B, C C {0,1}*. Prove: If A {0,1}* {0,1}* defined by (x, y) = 011|1xy for all x, y {0,1}* Given a language B C {0,1}*, define the language 3B C {0,1}* by 3B = {x {0,1}* |(Ew e {0,1}*)(x, W) E B}. Problems 50 and 51 characterize computable enumerability in terms of unbounded existential search. Problem 50. Let B C {0,1}*. Prove: If B is c.e., then 3B is c.e. Problem 51. Prove: For every c.e. language A C{0,1}*, there is a decidable language B C {0,1}* such that A= B. Recall that the join of two languages A, B C {0,1}* is AU B = {x0 | X A} U{y1|y E B}. Problem 52. Let A, B, C C {0,1}*. Prove: If A