6. Let L1 and L2 be two languages. (a) Define the exclusive-or (XOR) operation as Le La = {rl either r E Li orr e L2 but not both) Prove or disprove that Turing-recognizable languages are closed under the XOR Answer: operation. (b) Define the divide operator as L/L ( ry E L1 for some y e La) To illustrate the divide operator, suppose that Li (w w is a palindrome) and that L2 (wl w has even number of a's and b's). Since aababbbabaa E L and bbabaa E L2 then aabab E L/L2 because aabab + bbabaaaababbbabaa. Prove or disprove that Turing-recognizable languages are closed under the divide operation. Answer: 7. Let L = {(M,j) Af is a Turing machine that accepts at least one string in at most j steps.), Prove that L is decidable. Answer: 6. Let L1 and L2 be two languages. (a) Define the exclusive-or (XOR) operation as Le La = {rl either r E Li orr e L2 but not both) Prove or disprove that Turing-recognizable languages are closed under the XOR Answer: operation. (b) Define the divide operator as L/L ( ry E L1 for some y e La) To illustrate the divide operator, suppose that Li (w w is a palindrome) and that L2 (wl w has even number of a's and b's). Since aababbbabaa E L and bbabaa E L2 then aabab E L/L2 because aabab + bbabaaaababbbabaa. Prove or disprove that Turing-recognizable languages are closed under the divide operation. Answer: 7. Let L = {(M,j) Af is a Turing machine that accepts at least one string in at most j steps.), Prove that L is decidable