Math 2000: Set TheoryAssignment 3Due Date: Oct. 1 2012Problem 1.LetA, B, Cbe sets. Determine if the following assertions are true or false. Provide a proof orcounterexample in each case.a)(A∩B)⊂(B∩C)⇒A⊂Bb)(A∪B)⊂(B∪C)⇒A⊂BProof.a) False. Take as counterexampleA={a,1,2,3},B={b,1,2,3,4,5},C={c,1,2,3,4,5}. Workthrough why this fails. Of course there are other counterexamples,.b) False. Take as counterexampleA={1,2,3},B={a, b, c},C={0,1,2,3,4}. Work through why thisfails. Of course there are other counterexamples,.Problem 2.LetA, B, Cbe subsets ofU. ProveA∩(B∪C) = (A∩B)∪(A∩C).
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Problem 3.LetA,B, andCbe sets in some universeU.Prove thatA∩(BΔC) = (A∩B)Δ(A∩C)by showing thatA∩(BΔC)and(A∩B)Δ(A∩C)are subsets of each other.Proof.Supposex∈A∩(BΔC). Thenx∈Aandx∈BΔC, sox∈A,x∈(B∪C), andx /∈B∩C. So ifx∈Bthenx /∈C, in which casex∈A∩Bandx /∈A∩C, sox∈(A∩B)∪(A∩C