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T2.1 - A set is a collection of objects from a universe The...

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A set is a collection of objects from a universe. The objects are called members or elements . They are not in any order, despite how they may be written, nor may sets contain duplicate members. Sets may be denoted in several ways: (1). By enumeration, if finite. E.g. {1,2,3} (Same as {2,1,3} — same as {1,1,2,3}) (2). By enumeration with dots of ellipses, where obvious. {2,4,6 . .} (3). By use of a propositional function: {x | x>3} (Need to know universe!) Or in general, {x | P(x)} or “the set of all elements that make P true.” E.g. {x|x>3} = {4, 5, . .} in domain of integers 1
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Define: x S to mean “x is an element of S.” Note that x S is a proposition. We also define x S to mean –(x S) or “x is not an element of S.” Define |S| as “cardinality of S” or the number of elements in S. Define: S T as “S is a subset of T” S T iff 2200 x(x S -> x T) Define: S=T iff S and T contain the same elements, or iff 2200 x(x S <->x T) 2
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Theorem: S=T iff S T /\ T S Proof: S=T iff 2200 x(x S <->x T) iff 2200 x[(x S ->x T) /\ (x T ->x
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