assignment 2 solutions (Set Theory) - Math*2000 Set Theory Assignment 2 Due Date Sept 24 2012 Problem 1 Using the laws of the algebra of propositions

# assignment 2 solutions (Set Theory) - Math*2000 Set Theory...

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Math*2000: Set Theory Assignment 2 Due Date: Sept. 24 2012 Problem 1.Using the laws of the algebra of propositions and the rules of inference prove thatPQ, RS,¬(QS), R‘ ¬Pis a valid argument. Proof. 1. P Q 2. R S 3. ¬ ( Q S ) 4. R ————– 5a. S (from 4 and 2 by Modus Ponens) 5b. ¬¬ S (from 5a by the Involution law)(it’s okay if this step isn’t explicitly listed) 6. ¬ Q ∨ ¬ S (from 3 by DeMorgan’s) 7. ¬ Q (from 5b and 6 by Disjunctive Syllogism) 8. ¬ P (from 1 and 7 by Modus Tollens) (or, from 1 and 7 by the truth table for implication) Problem 2.Negate the following propositions.1. For everyxR, there is a characteristic functionχx.2.xy[xy= 1x6= 0]Proof. Problem 3.Indicate “true” or “false” in each of the following assertions.Explain.1.∅ ∈ {∅,{∅}}.2.∅ ⊂ {∅,{∅}}.3.{∅} ⊂ {∅,{∅}}.4.{∅} ∈ {∅,{∅}}.5.{{∅}} ⊂ {∅,{∅}}.Proof. 5. True.{∅} ∈ {{∅}} ⇒ {∅} ∈ {∅,{∅}}, so by definition of a subset,{{∅}} ⊂ {∅,{∅}}.
Problem 4.Calculate/Determine: 1.{∅}Δ{∅,{∅}}2.{∅,{∅}}Δ{∅,{∅},{∅,{∅}}}3.{∅}Δ{∅,{∅},{∅,{∅}}} 4. Are any of the setsA={∅}Δ{∅,{∅}},B={∅,{∅}}Δ{∅,{∅},{∅,{∅}}}, andC={∅}Δ{∅,{∅},{∅,{∅}}}subsets of one another? If so, indicate whether or not the subset is proper. Proof. 1. We find{∅}Δ{∅,{∅}}=({∅} ∪ {∅,{∅}})\({∅} ∩ {∅,{∅}})={∅,{∅}}\{∅}={{∅}}.2. We find{∅,{∅}}Δ{∅,{∅},{∅,{∅}}}=({∅,{∅}} ∪ {∅,{∅},{∅,{∅}}})\({∅,{∅}} ∩ {∅,{∅},{∅,{∅}}})={∅,{∅},{∅,{∅}}}\{∅,{∅}}={{∅,{∅}}}.3. We find{∅}Δ{∅,{∅},{∅,{∅}}}=({∅} ∪ {∅,{∅},{∅,{∅}}})\({∅} ∩ {∅,{∅},{∅,{∅}}})={∅,{∅},{∅,{∅}}}\{∅}={{∅},{∅,{∅}}}.4. We seeA(Cbecause the only element{∅} ∈Ashows up as the first element ofC, andB(C, asthe only element{∅,{∅}}inBshows up as the second element ofC.
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