Introduction to Set Theory with python

A set is a collection of objects. For instance, let's say that 1, 2, 3 ,4, a ,b ,c , $ are objects. A set of those objects would be

S = {1, 2, 3, 4, a, b, c, $}

#python code
S = {1, 2, 3, 4, 'a', 'b', 'c', '$'}

#It is also possible to use the set() constructor to build a set
S = set((1, 2, 3, 4, 'a', 'b', 'c', '$'))

# Either way, you can make use of the methods:
S.update({'y', 'z'}) # Adds multiple elements to the set
S.add('w') # Adds one element to the set
S.remove('y') # Removes the element from the set
S.remove('z')
S.remove('w')

To set are equal if they contain the same objects. Order and repetition does not matter.

{1, 2, 3, 4, a, b, c, $} is equal to {a, b, c, 1, 2, 3, 4, $, $, $}

#python code
V = {'a', 'b', 'c', 1, 2, 3, 4, '$', '$', '$'}
print(S == V) # Output is True

#If we print V we will notice that the output does not include repetitions

print(V) #output {1, 2, 3, 4, 'a', '$', 'c', 'b'}

If, S = {1, 2, 3, 4, a, b, c, $}

a ∈ S, means object a is an element of the set S.

x ∉ S, means object s is not an element of the set S.

#python code
print('a' in S) #Output is True
print('x' in S) #Output is False

The number of unique elements in a set is called Cardinality.

The cardinality of S is 8.

|S| = 8

#python code
print(len(S)) #Output is 8

Set equivalence is when two or more sets have the same cardinality.

S is equivalent to V.

#python code
def areSetsEquivalent(setA, setB):
    if len(setA) == len(setB):
        return True
    else:
        return False

print(areSetsEquivalent(S, V)) # Output is true

---

Operations

Cartisian Product

In Set Theory, there is an operation called Cartisian product.

It is denoted as:
S x V;

And read as:
S cross V.

It is the set of all ordered pairs (a, b) where a is in A and b is in B.

A x B = { (a,b) | a ∈ A and b ∈ B}

{a1, a2} x {b1, b2} = { (a1, b1), (a1, b2), (a2, b1), (a2, b2) }

Example

A = {1 , 2}
B = {3, 4}

A x B = { (1, 3), (1, 4), (2, 3), (2, 4) }

The cardinality of the Cartesian product is equal to the product of the Cardinality of the sets.

|A x B| = |A| * |B|;

|A x B| = 2 * 2;

|A x B| = 4;

#python code
from itertools import product
A = {1 , 2}
B = {3, 4}
print(set(product(A, B))) #Output {(2, 3), (2, 4), (1, 3), (1, 4)}

Union

The union of a collection of sets is a set that contains all the elements of those sets.

A ∪ B = {x | x ∈ A or x ∈ B}

Example:

A = {1 , 2}
B = {3, 4}

A ∪ B = {1, 2, 3, 4}

#python code
A = {1 , 2}
B = {3, 4}

print(A.union(B)) # Output {1, 2, 3, 4}

Intersection

The intersection of sets are all those objects that are common on all the sets.

A ∩ B = {x | x ∈ A and x ∈ B}

Example:

Y = {1, 2, 3}
Z = {3, 4, 5}

Y ∩ Z = {3}

#python code
Y = {1, 2, 3}
Z = {3, 4, 5}

print(Y.intersection(Z)) #Output {3}

Conclusion

As you can see, python provide a lot of tools to work with sets. The topics exposed on the content above are just an introduction to the Set theory. There is a lot more related the topic, such as, Difference and complement, Disjoint sets, Naturally Disjoint sets, partition of sets, subsets and power sets. I would discuss all of those topics in different posts.

Written by: E. A. Arroyo.