let 5 be the set at all strings in a's and b's, and define C: S 5by (Cis called ca:alenalian by a un the le ) (a) Prove that Cis one to one. Proof: Suppose s, and sz are strings In 3 such thet Cts) - Cts). Use the definldon of C'to write this ceuetdon in terms of a, 5, and s, as fullows: Now strings are finite socuences ot characters, and since the strings on bath sidcs at the above equation are equal, tor cach Intogarn2 0, the nth character trom the ict in tihe latt-hand string ---Seect- - the oth character trom the let in the nght-hand strina. It follows that for cach Integer n0, the oth character from the let in s, oquals thc nth character from the let In s.. Hence s, - 5, and so C: one-to-cne. (b) Give a oaunterexample to show that C is not onta. Counterexample: The strina k in 5 but is not ecual to Cs) tor any string s harausa evary string in the range at C starts with

let 5 be the set at all strings in a's and b's, and define C: S 5by (Cis called ca:alenalian by a un the le ) (a) Prove that Cis one to one. Proof: Suppose s, and sz are strings In 3 such thet Cts) - Cts). Use the definldon of C'to write this ceuetdon in terms of a, 5, and s, as fullows: Now strings are finite socuences ot characters, and since the strings on bath sidcs at the above equation are equal, tor cach Intogarn2 0, the nth character trom the ict in tihe latt-hand string ---Seect- - the oth character trom the let in the nght-hand strina. It follows that for cach Integer n0, the oth character from the let in s, oquals thc nth character from the let In s.. Hence s, - 5, and so C: one-to-cne. (b) Give a oaunterexample to show that C is not onta. Counterexample: The strina k in 5 but is not ecual to Cs) tor any string s harausa evary string in the range at C starts with

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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