8.2: set operations
- Page ID
- 26495
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Sets are not only powerful due to their ability to maintain unique elements but also because they offer a suite of operations that can perform complex manipulations with succinct and readable code. These operations, borrowing concepts from mathematical set theory, include unions, intersections, differences, and symmetric differences. Here are some key methods and operators you can use with sets:
Union
The union of two or more sets is a fundamental operation that combines all elements from the involved sets, ensuring that each element is unique. In Python, the union can be performed using the `union()` method or the `|` operator.
set_a = {1, 2, 3}
set_b = {3, 4, 5}
union_set = set_a.union(set_b) # or set_a | set_b
print(union_set) # Output: {1, 2, 3, 4, 5}
Note that 3 appears in both sets, but only appears once in the union. The union operation is particularly useful when you need to gather all distinct items from multiple collections.
Intersection
Intersection creates a new set containing only the elements that are present in both sets. This operation is useful for finding commonalities between datasets. Similar to union, intersection can be achieved with the `intersection()` method or the `&` operator.
set_a = {1, 2, 3}
set_b = {3, 4, 5}
intersection_set = set_a.intersection(set_b) # or set_a & set_b
print(intersection_set) # Output: {3}
Intersections are often used in filtering processes where items meeting a certain criteria across multiple sets are required.
Difference
The difference operation yields a set containing elements that are in one set but not in the others. It's performed using the `difference()` method or the `-` operator. This is particularly useful for identifying elements that are unique to one dataset.
set_a = {1, 2, 3}
set_b = {3, 4, 5}
difference_set = set_a.difference(set_b) # or set_a - set_b
print(difference_set) # Output: {1, 2}
This operation can be used to exclude specific items from a set based on the contents of another.
Symmetric Difference
Symmetric difference produces a set with elements that are in either of the sets but not in both. It can be executed with the `symmetric_difference()` method or the `^` operator. This operation is useful for finding items that are unique to each set when comparing two datasets.
set_a = {1, 2, 3}
set_b = {3, 4, 5}
symmetric_difference_set = set_a.symmetric_difference(set_b) # or set_a ^ set_b
print(symmetric_difference_set) # Output: {1, 2, 4, 5}