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am_lect_20 - Inductively Defined Sets We can define sets...

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Inductively Defined Sets We can define sets inductively/recursively as well. Example 1: The set S is defined as 0 S n S , n + 1 S Note that S = N (the set of natural numbers) Example 2: The set T is defined as (0, 3) T ( w , x ) T , ( y , z ) T , ( w + y , x + z) T ( w , x ) T , ( y , z ) T , ( w y , x z) T Note that T = { (0, 3 m ) m Z }
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Comparing two functions Which of these two procedures is faster? 1: procedure NAE1(a : array of n reals) : bool 2: for i := 1 to n-1 3: for j := i+1 to n 4: if a[i] a[j] 5: return true 6: return false 1: procedure NAE2(a : array of n reals) : bool 2: for i := 1 to n-1 3: if a[i] a[i+1] 4: return true 5: return false Convince yourself that both procedures do the same thing. Suppose NAE1 takes T 1 ( n ) steps and NAE2 takes T 2 ( n ) steps on an array of size n It is clear that n N , T 1 ( n ) T 2 ( n ), so NAE1 is “faster”. We will
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