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

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Math*2000: Set TheoryAssignment 2Due Date: Sept. 24 2012Problem 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.PQ2.RS3.¬(QS)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.1. There existsxRthat does not have a characteristic functionχx.2. The negation isxy¬[xy= 1x6= 0], which we can simplify as:xy,[xy6= 1x= 0]Problem 3.Indicate “true” or “false” in each of the following assertions.Explain.1.∅ ∈ {∅,{∅}}.2.∅ ⊂ {∅,{∅}}.3.{∅} ⊂ {∅,{∅}}.4.{∅} ∈ {∅,{∅}}.5.{{∅}} ⊂ {∅,{∅}}.Proof.1. True as the empty set is listed as the first element of this set.2. True. In class we saw the empty set is always a subset of any set.3. True:∅ ∈ {∅} ⇒ ∅ ∈ {∅,{∅}}, so by definition of a subset,{∅} ⊂ {∅,{∅}}.4. True as{∅}is listed as the second element in the given set.
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