4 - is contained in { 1 , 2 , 5 } . Since there are no...

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The function h takes a list of integers and an integer n , and returns a sublist of elements less than n . 2.1.1 Subsets D EFINITION 2.2 (S UBSETS ) Let A , B be any two sets, Then A is a subset of B , written A B , if and only if all the elements of A are also elements of B : that is, A B ⇔ ∀ objects x. ( x A x B ) We have written the subset deFnition in two ways, using English and also using logical notation which will be familiar from the lectures on logic. Ei- ther style is appropriate for this course. Just use whichever suits you best. Notice that the deFnition talks about all objects x , although we have not said what an object is! We assume that there is an underlying universe of discourse when discussing sets: that is, the set of all possible objects under discussion. This set is sometimes written U . Any set is a subset of itself. Other simple examples are { a,b } { a,b,c } { c, c,b } { a,b, c,d } N Z ∅ ⊆ { 1 , 2 , 5 } This last example is tricky. To convince ourselves that it is true we need to show that every element in
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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 convinc-ing 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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