hw_00_sol - Problem Set 0 Spring 2010 Due Thursday Jan 28...

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Unformatted text preview: Problem Set 0 Spring 2010 Due: Thursday Jan 28, 2:00pm, in class before the lecture. Please follow the homework format guidelines posted on the class web page: http://www.cs.uiuc.edu/class/sp10/cs373/ 1. [ Category : Notation, Points : 20] Answer each of the following my marking each with true , false , or wrong nota- tion . Follow the notations in Sipser . { ... } is used to represent sets and not multisets or anything else. D1) { a,b,c } ∩ { d,e } = {} D2) { a,b,c } ∩ { d,e } = { ∅ } D3) { a,b,c } ∪ { d,a,e } = { a,b,c,d,a,e } D4) { a,b,c } ∪ { d,a,e } = { a,b,c,d,e } D5) { a,b,c } \ { a,d } = { b,c } D6) ∅ ∈ { ∅ ,a,b,c } D7) ∅ ⊆ { ∅ ,a,b,c } D8) ∅ ∈ ∅ D9) a ⊆ { ∅ ,a,b,c } D10) { a,c } + { c,b } = { a,b,c } D11) { a,b } - { b } = { a } D12) { a,a } = { a } D13) {{ a } , { a }} = { a,a } D14) a ∈ { a, { a } , {{ a }}} D15) { a } ∈ { a, { a } , {{ a }}} D16) {{{ a }}} ⊆ { a, { a } , {{ a }}} D17) { ∅ } = {{}} D18) { a,b } × { c,d } = { ( a,c ) , ( b,d ) } D19) { a,b } × { c,d } = { c,d } × { a,b } D20) |{ a,b } × { a,b }| = 3 Solution: D1) { a,b,c } ∩ { d,e } = {} true D2) { a,b,c } ∩ { d,e } = { ∅ } false 1 D3) { a,b,c } ∪ { d,a,e } = { a,b,c,d,a,e } true D4) { a,b,c } ∪ { d,a,e } = { a,b,c,d,e } true D5) { a,b,c } \ { a,d } = { b,c } true D6) ∅ ∈ { ∅ ,a,b,c } true D7) ∅ ⊆ { ∅ ,a,b,c } true D8) ∅ ∈ ∅ false D9) a ⊆ { ∅ ,a,b,c } wrong notation D10) { a,c } + { c,b } = { a,b,c } wrong notation D11) { a,b } - { b } = { a } wrong notation (but we will also accept "true") D12) { a,a } = { a } true D13) {{ a } , { a }} = { a,a } false D14) a ∈ { a, { a } , {{ a }}} true D15) { a } ∈ { a, { a } , {{ a }}} true D16) {{{ a }}} ⊆ { a, { a } , {{ a }}} true D17) { ∅ } = {{}} true D18) { a,b } × { c,d } = { ( a,c ) , ( b,d ) } false D19) { a,b } × { c,d } = { c,d } × { a,b } false D20) |{ a,b } × { a,b }| = 3 false 2. [ Category : Proof, Points : 16] Professor Moriarty claims that he has a way of describing every real number between 0 and 1 using an English sentence (of nite length), i.e. for every real number r , there is an English sentence s that precisely describes r . Prove that Professor Moriarty is wrong. Note: Assume that a real number between 0 and 1 is of the form .a 1 a 2 a 3 ... , where each a i ∈ { , 1 ,... 9 } , i.e. is an in nite set of decimal points. This is not quite true, as . 09999999 ... is actually the same as . 10000 ... , but ignore this subtlely for this question....
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