09-14-review-1 - b x ∈ ∪F ∩G here F and G are...

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Sets and Logic — Fall 2011 Wednesday, 9/14/11 A Few Review Problems for Midterm #1 1 Analyze the logical forms of the following quotes. a. “All you need is ignorance and conFdence and the success is sure.” — Mark Twain. b. “I’ve never met a man I didn’t like” — Will Rogers. c. “±or every moment of triumph, for every instance of beauty, many souls must be trampled.” — Hunter S. Thompson. (Don’t worry about translating the “many”.) d. “Always do everything you ask of those you command.” George S. Patton. (Use a proposition C ( x ) for “you command x ” and A ( x,y ) for “you ask x to do y ”.) 2 Analyze the logical forms of the following statements. Your answers should not use , n , P , ∩ ∪ , \ , { , } or ¬ . a. F ⊆ P ( A ); here F is a family of sets, while A is a set.
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Unformatted text preview: b. x ∈ ∪F \ ∩G ; here F and G are families of sets. c. x ∈ u i ∈ I ( A i ∩ B i ); here each A i and B i is a set, and I is an index set. d. x ∈ ∪{P ( A ) | A ∈ F} ; here F is a family of sets. 3 Show that ( ∩F ) ∩ ( ∩G ) = ∩ ( F ∪G ); here F and G are families of sets. 4 Let us deFne A = ∅ . Then for each n ≥ 1, let A n = A n-1 ∪ { A n-1 } . Please don’t freak out: this is not a potential exam problem (yet). a. Write out A 1 , A 2 , and A 3 . b. What is | A n | ? c. DeFne A ω = u n ∈ N A n . What is | A ω | ? d. Make a sensible deFnition for A ω +1 . What is | A ω +1 | ? Is this set equal to A ω ? Why or why not? e. Make a sensible deFnition for A 2 ω ....
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