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5.2.9 - sets like in constructed from unions intersections...

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INFS 501 - Problem 5.2.9 Suppose A, B, and C are sets. Show : A B  ∩ C B  =  A C  − B. (*) Solution 1 : The textbook solution is a logical but lengthy and, for the novice, a confusing 2-step process: 1st, show the set on the left hand side (LHS) of (*) is a subset of the set on the RHS of (*). 2nd, show the RHS set is a subset of the LHS set. To show each required subset relationship, start with a variable x belonging the purported subset & show that x must necessarily belong to the purported superset. ... Comment : I recommend the more practical and intuitive Venn Diagram approach. A Venn diagram may be used to decipher any
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Unformatted text preview: sets like in (*), constructed from unions, intersections, and set-complements. Solution 2 : Starting with the 3 sets in this example, the Venn diagram has 2 3 = 8 regions: R1, . .. R8. See below. From the diagram, the set on the LHS of (*) consists of the regions { R1, R4 }∩{ R4,R7 }={ R4 } . The set on the RHS of (*) consists of the regions { R4, R5 }− { R2,R3,R5,R6 }= { R4 } . Thus, the sets on the LHS and RHS both consist exactly of the same region, namely R4. Thus, the LHS & RHS sets are the same. A B C R1 R2 R3 R4 R5 R6 R7 R8...
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