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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
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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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
Transcribed Image Text: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
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