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Intersection[list1, list2] {5, 7} Complement[universe, list1] {2, 4, 6, 8, 9, 10} Complement[universe, list1, list2] {2, 4, 6, 9}
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Using the Basic Math Input palette, the symbols and may be used to represent union and intersection, respectively.
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list1 list2 is equivalent to Union[list1, list2]. list1 list2 is equivalent to Intersection[list1, list2].
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list1 = {1, 2, 3, 4, 5}; list2 = {3, 4, 5, 6, 7};
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list1 list2 2, 3, 4, 5, 6, 7}
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list1 list2 {3, 4, 5}
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A subset of A is any set, each of whose elements are members of A. The empty set is a subset of every set. Including the empty set, a set of n elements has 2n subsets. The set of all subsets of A is called the power set of A.
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Subsets[list] returns a list containing all subsets of list, including the empty set, i.e., the power set of list.
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There are a number of useful set commands available in the package Combinatorica`. Among them are CartesianProduct and KSubsets. By definition, the Cartesian product of two sets, A and B, is the set of ordered pairs of elements, the first taken from A and the second from B.
CartesianProduct[list1, list2] returns the Cartesian product of list1 and list2. KSubsets[list, k] returns a list containing all subsets of list of size k.
EXAMPLE 40
Combinatorica` list1 = {a, b, c, d}; list2 = {x, y, z};
{{a,
This loads the package. See 1.
CartesianProduct[list1, list2] x}, {a, y}, {a, z}, {b, x}, {b, y}, {b, z}, {c, z}, {d, x}, {d, y}, {d, z}}
x}, {c, y},
EXAMPLE 41
list = {a, b, c, d}; Subsets[list]
{{}, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d}}
Combinatorica`
Omit if you have already loaded the package.
KSubsets[list, 3] {{a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}}
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SOLVED PROBLEMS
3.24 Which distinct letters are contained in the word MISSISSIPPI (Compare with Problem 3.18)
SOLUTION
Union[Characters["MISSISSIPPI"]] {I, M, P, S}
3.25 Find the union and intersection of the sets {a, b, c, d, e, f, g}, {c, d, e, f, g, h, i}, and {e, f, g, h, i, j, k}.
SOLUTION
set1 = {a, b, c, d, e, f, g}; set2 = {c, d, e, f, g, h, i}; set3 = {e, f, g, h, i, j, k};
Union[set1, set2, set3] or set1 set2 set3 b, c, d, e, f, g, h, i, j, k}
Intersection[set1, set2, set3] or set1 set2 set3 f, g}
3.26 Find all the elements of the set {a, b, c, d, e, f, g} that are not in {a, c, d, e}.
SOLUTION
universe = {a, b, c, d, e, f, g}; set = {a, c, d, e};
Complement[universe, set] f, g}
3.27 The 20th prime is 71. Find all the numbers not exceeding 71 that are not prime.
SOLUTION
universe = Range[71]; primes = Prime[Range[20]]
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71}
Complement[universe, primes] {1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70}
3.28 Construct a list consisting of the consonants of the alphabet.
SOLUTION
letters = CharacterRange["a", "z"]; vowels = Characters["aeiou"]; consonants = Complement[letters, vowels]
c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}
3.29 Find all the numbers less than 1000 that are both prime and Fibonacci.
SOLUTION
k = 1; list1 = {}; While[Fibonacci[k] 1000, list1 = Append[list1, Fibonacci[k]]; k++]
Lists
k = 1; list2 = {}; While[Prime[k] 1000, list2 = Append[list2, Prime[k]]; k++]
list1 list2 3, 5, 13, 89, 233}
3.30 Create a list that contains all the subsets of {a, b, c, d, e}. How many subsets are there
SOLUTION
letters = {a, b, c, d, e}; Subsets[letters]
{{}, {a}, {b}, {c}, {d}, {e}, {a, b}, {a, c}, {a, d}, {a, e}, {b, c}, {b, d}, {b, e}, {a, d, e}, {b, c, d}, {b, c, e}, {b, d, e}, {c, d, e}, {a,
d}, {c, e}, {d, e}, {a, b, c}, {a, b, d},{a, b, e}, {a, c, d}, {a, c, e}, b, c, d}, {a, b, c, e}, {a, b, d, e},{a, c, d, e}, {b, c, d, e}, {a, b, c, d, e}}
Length[%] 32
3.31 Create a list of all the subsets of {a, b, c, d, e} that contain precisely three elements. How many are there
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