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Exercise 1 for strings and sets

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1. What does each of the following expressions give?

(a) a5 (b) |aba7b| (c) || (d) xR , where x = abab

(e) AB, where A = {a, b, c}, B = {aaaa, bbbb, cccc}.

(f) A*BA* ,where A and B are the same sets defined in (e) above.

Answer: (a) aaaaa (b) 10 (c ) 0 (d) baba

(e) AB = {aaaaa, abbbb, acccc, baaaa, bbbbb, bcccc, caaaa, cbbbb, ccccc}

(f) A*BA* = { x | x = {a, b, c}* and x has aaaa, bbbb, or cccc as a substring}

2. For each of the following sets, which of the strings given below in (1) – (7) are its members?

(a) {xyxR | x {a, b, c}*, y ccc } (b) {xx | x {a, b, c}* }

(c) {x | x {a, b, c}* and x has more a’s than b’s. } {aibj | j, i > 0 }

(d) ({a, b, c}* - ({aibj | i > j > 0 } {aibj | 0 < i < j })) {aibj | j, i > 0 }

(1) aaaabbbb (2) aaaa (3) aaaacccaaaa (4) bbbaaaa (5) abcccccba (6) aaaaab (7) abaaba

Answer: (a): (3), (5) (b): (2), (7)

(c): (6) (Notice that the members should be in {aibj | i > j > 0})

(d): (1) (Notice that the members should be in {aibi | i > 0 })