John Perry. Fall 2009
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1 Lecture 5: Sage University of Southern Mississippi Fall 2009
2 Outline You should be in worksheet mode to repeat the examples.
3 Collections? Collection: group of objects identified as single object ordered points x 0, y 0, x0, y 0, z 0 tuples a 0, a 1, a 2,...a n lists a 0, a 1,...,a n sequences a 0, a 1, a 2,... unordered sets {a 0, a 5, a 3, a 2, a 1 }
4 Outline
5 collections Sage offers several collections standard in ordered ( sequence types ) tuples, lists access ith element using [i-1] unordered ( set types ) sets cannot access ith element only one instance of any element
6 Tuples A tuple is an immutable, ordered collection immutable: cannot change elements defined using parentheses
7 Example sage: my_tuple = (1,5,0,5) 4-tuple sage: my_tuple[2] access 3rd element 0 sage: my_tuple[2] = 1 assign to 3rd element?... Output deleted... TypeError: 'tuple' object does not support item assignment sage: (1,5,0,5) my_tuple
8 Lists A list is a mutable, ordered collection mutable: can change elements defined using square brackets
9 Example sage: my_list = [1,5,0,5] list of 4 elements sage: my_list[2] access 3rd element 0 sage: my_list[2] = 1 assign to 3rd element? sage: my_list[2] 1 no error! access gives new value! sage: [1,5,1,5] my_list
10 Sets A set is a mutable, unordered collection defined using set(tuple or list) define empty set using set() redundant elements automatically deleted elements may not remain in the order you supply them
11 Example sage: my_set = set([1,5,0,5]) set of 4 elements sage: my_set[2] access 3rd element?... Output deleted... TypeError: 'set' object is unindexable sage: my_set so what s in there, anyway? set([0, 1, 5]) not original list!
12 Does he do any tricks? (1) sets, tuples, and lists len(c) number of elements in C x in C is x an element of C? tuples and lists C.count(x) Number of times x appears in C C.index(x) First location of x in C C1 + C2 join C1 to C2, returned as new tuple/list
13 Example sage: len(my_set) 3 sage: 4 in my_set False sage: 5 in my_set True sage: my_tuple.count(5) 2 How many 5s? sage: my_list.index(5) 1 in second location sage: my_list + [1,3,5] [1, 5, 0, 5, 1, 3, 5]
14 lists Does he do any tricks? (2) L.append(x) adds x to the end of L L.extend(C) appends each element of the collection C L.insert(i, x) insert x at L[i], shifting this and subsequent elements back L.pop(i) delete L[i] and tell me its value L.remove(x) remove first instance of x from L L.reverse() reverse the order of elements L.sort() sort the elements of L according to its natural order only a good idea for primitive elements
15 Example sage: my_list [1, 5, 0, 5] sage: my_list.extend((2,4)) sage: my_list [1, 5, 0, 5, 2, 4] sage: my_list.insert(3,-1) sage: my_list [1, 5, 0, -1, 5, 2, 4] sage: my_list.pop(3) -1 sage: my_list.sort() sage: my_list [0, 1, 2, 4, 5, 5]
16 start: A word on inserting my_list L[0] L[1] L[2] L[3] L[4] L[5] sage: my_list.insert(3,-1)
17 start: A word on inserting my_list L[0] L[1] L[2] L[3] L[4] L[5] sage: my_list.insert(3,-1)
18 start: A word on inserting my_list L[0] L[1] L[2] L[3] L[4] L[5] sage: my_list.insert(3,-1) my_list L[0] L[1] L[2] L[3] L[4] L[5] L[6]
19 Does he do any tricks? (3) sets S.add(x) adds x to S S.clear() removes all elements from S S.difference(C), S.intersection(C), S.union(C) set operations: difference S\C, intersection S C, union S C S.isdisjoint(C) True iff S and C share no elements S.pop() removes and reports random (first?) element of S S.remove(x) remove x from S S.symmetric_difference(x) set operation: symmetric difference S\C C\S
20 Example sage: sage: sage: my_set = set([1,5,0,5]) my_set.add(4) my_set set([0, 1, 4, 5]) sage: False sage: my_set.isdisjoint([-1,-2,4]) my_set.symmetric_difference([-1,-2,4]) set([-2, -1, 0, 1, 5]) sage: my_set.remove(2)... Output removed... KeyError: 2 sage: sage: [0, 4, 5] my_set.remove(1) my_set
21 Outline
22 Negative indices have meaning: Tricks with [] L[0] L[1] L[2] L[3] L[4] IndexError IndexError L[-5] L[-4] L[-3] L[-2] L[-1]
23 Negative indices have meaning: Tricks with [] L[0] L[1] L[2] L[3] L[4] IndexError IndexError L[-5] L[-4] L[-3] L[-2] L[-1] Example sage: L = [1,5,0,5] sage: 5 sage: 1 L[-1] L[-4] sage: L[-5]... Output deleted... IndexError: list index out of range
24 Tricks with [:] C[first:last+1] specifies subcollection C[first] C[first+1]... C[last] omit first? = start at C[0] omit last? = end at C[-1]
25 Tricks with [:] C[first:last+1] specifies subcollection C[first] C[first+1]... C[last] omit first? = start at C[0] omit last? = end at C[-1] Example sage: L[2:4] L[2] to L[3] [0, 5] sage: L[:2] L[0] to L[1] [1,5] sage: L[2:] L[2] to L[-1] [0,5] sage: L[:] L[0] to L[-1] [1,5,0,5]
26 The range() command range(first,last+1) generates a list with n = last first elements where first is the first integer in the list default value is 0 last is the last integer in the list first last? empty list
27 Example sage: range(5) [0, 1, 2, 3, 4] sage: range(1,5) [1, 2, 3, 4] sage: range(3,5) [3,4] sage: range(5,5) [] sage: range(6,5) []
28 Outline
29 String: ordered collection of characters Hello H e l l o extract elements using [] join elements using + other useful functions on pg. 96 of text
30 Example sage: name = 'Euler' sage: name[2] 3rd character 'l' sage: name[-1] last character 'r' sage: name[0:4] first four characters in string 'Eule' sage: name + ' computed' add string; notice space 'Euler computed'
31 The str() command str(x) where x is any object that can be turned into a string Sage will turn a lot of objects into strings!
32 Example Numbers: sage: name + ' computed' + ' e**(i*pi) + 1 = ' + str(0) 'Euler computed e**(i*pi) + 1 = 0'
33 Example Numbers: sage: name + ' computed' + ' e**(i*pi) + 1 = ' + str(0) 'Euler computed e**(i*pi) + 1 = 0' Equations: (after obvious simplifications!) sage: name + ' computed ' + str(e**(i*pi) + 1 == 0) 'Euler computed 0 == 0'
34 Outline
35 Through, Sage offers several kinds of collections tuples, lists, sets Operations [] for extraction negatives allowed [:] gives subcollections usual mathematical operations on sets others supplied by allow lists of characters str(x) produces obvious string representation of x
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