Math*2000: Set TheoryAssignment 2Due Date: Sept. 24 2012Problem 1.Using the laws of the algebra of propositions and the rules of inference prove thatP⇒Q, R⇒S,¬(Q∧S), R‘ ¬Pis a valid argument.Proof.1.P⇒Q2.R⇒S3.¬(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 everyx∈R, there is a characteristic functionχx.2.∀x∀y[xy= 1∧x6= 0]Proof.1. There existsx∈Rthat does not have a characteristic functionχx.2. The negation is∃x∃y¬[xy= 1∧x6= 0], which we can simplify as:∃x∃y,[xy6= 1∨x= 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.