Unformatted text preview: ∅ is contained in { 1 , 2 , 5 } . Since there are no elements in ∅ , this property is vacuously true: ∅ is a subset of every set! P ROPOSITION 2.3 Let A , B , C be arbitrary sets. If A ⊆ B and B ⊆ C then A ⊆ C . Notice that we have stated that this property holds for arbitrary (all) sets A , B and C : such properties are universal properties on sets, in the sense that they are hold for all sets. What does it mean to convince ourselves that such properties are true? In the logic course, you have been exploring very formal deFnitions of a proof within a logical system. In this course, we do not have to be quite so formal. However, the proofs should be convincing even though they are not written within a purely logical setting. When faced with constructing a proof, check three things: (1) that the arguments put forward are all true and the sequence follows logically from beginning 5...
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This note was uploaded on 01/02/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '09 term at SUNY Empire State.
 Spring '09
 Koskesh
 Math, Integers, Sets

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