Math 230 EFall 2011Homework 2Drew ArmstrongProblem 1.Letx⊕ydenote the XOR (exclusive or) function, which is defined by:1⊕1 = 0,1⊕0 = 1,0⊕1 = 1,0⊕0 = 0.Expressx⊕yin terms of the basic operations∧,∨,¬.Expressthe usual “inclusive or”x∨yin terms of the operations∧,⊕,¬.[Boolean logic could be based on either of the triples(∧,∨,¬)or(∧,⊕,¬). Once upon a timesomebody made a choice.]Problem 2.Given two subsetsS, T⊆Uof some universal set, the statement “S=T”means exactly that “∀x∈U, x∈S⇔x∈T”. Define thesymmetric differencebySΔT:=(S∩Tc)∪(T∩Sc).Verifythe following logical equivalence:“S=T” = “SΔT=∅”.(Hint: Use the principle¬(∀x∈U, P(x)) = (∃x∈U,¬P(x)) and de Morgan’s Law.)[Food for thought: What is the relationship betweenx⊕yandSΔT?]Problem 3.For this problem you may assume de Morgan’s Laws:¬(P1∧P2∧· · ·∧Pk) =¬P1∨¬P2∨· · ·∨¬Pk¬(P
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