Superset – Python

Subset and Superset are two fundamental concepts in set theory. Python also has built-in methods to support these concepts.

Supersets

A set is defined as the superset of another set, if second set all elements are also elements of first set. i.e. set A will be the superset of set B if all elements of set B are also elements of set A. In other words superset can be defined as inverse of subset.

Superset – Mathematical Definition

Superset is denoted with symbol ⊇. Above definition can be denoted as A ⊇ B. If set A is a superset of set B, then it means that set A and set B are either equal OR all elements of set B are also elements of set A.

In Mathematics, subset can be defined as A ⊇ B ⟺ ∀ x ( x ∈ B ⟹ x ∈ A ). Lets understand this with an example:

Superset symbol ⊇ also includes equality in it. This leads to the concept of Proper Superset.

Proper Superset: set A is a proper superset of set B, if A ⊇ B and A ≠ B. This is denoted as A ⊃ B. Example – A = {10, 20, 30} and B = {10, 20}. Here, A ⊃ B because set B is contained within set A and A ≠ B.

Properties of Supersets

• Reflexivity: Every set is always a superset of itself, A ⊇ A.
• Transitivity: If A ⊇ B and B ⊇ C, then A ⊇ C. It means, if set A is a superset of set B and set B is a superset of set C, then it make set A superset of Set C also. Example – A = {10, 20, 30, 40}, B = {10, 20, 30}, C = {10, 20}
• Antisymmetry: If A ⊇ B and B ⊇ A, then A = B. It means, if set A is a superset of set B and set B is a superset of set A, then both set A and set B should be equal. Example – A = {10, 20}, B = {10, 20}

Python provides built-in method issuperset() to check if the set is superset to another set or not. Lets see, how issuperset() method works in Python.

Using issuperset() method

• Syntax – set_1.issuperset(set_2)
• issuperset() method takes a single “iterable” as an argument.
• set_1.issuperset(set_2) method returns boolean (True or False) value. True means set_1 is a superset of set_2 and False means set_1 is not a superset of set_2.

# using issuperset() method
print("using issuperset() method")
set_1 = {10, 20, 30, 70, 80, 90}
set_2 = {70, 80, 90}

superset_set12 = set_1.issuperset(set_2)         # check set_1 is superset of set_2
superset_set21 = set_2.issuperset(set_1)         # check set_2 is superset of set_1

print(set_1)
print(set_2)
print(superset_set12)       # will return True = set_1 is superset of set_2
print(superset_set21)       # will return False = set_2 is not superset of set_1

We have defined two sets referenced by variables set_1 and set_2. Using built-in issuperset() method we are checking if set_1 and set_2 are superset to each other or not. Output from issuperset() method will be a boolean value.

Program Output

using issuperset() method
{80, 20, 90, 70, 10, 30}         # elements of set_1
{80, 90, 70}                            # elements of set_2
True                     # set_1 is superset of set_2
False                   # set_2 is superset of set_1

From the program output, set_1 and set_2 elements are printed. issuperset() method has return a boolean (True or False) value. True for set_1 is superset of set_2 and False when set_2 is not superset of set_1.

Using issuperset() method – with empty sets

• Syntax – set_1.issuperset(set_2)

# using issuperset() method - with empty set
print("using issuperset() method - with empty set")
set_1 = set()
set_2 = set()

superset_set12 = set_1.issuperset(set_2)         # check set_1 is superset of set_2
superset_set21 = set_2.issuperset(set_1)         # check set_2 is superset of set_1

print(set_1)
print(set_2)
print(superset_set12)       # will return True = set_1 is superset of set_2
print(superset_set21)       # will return True = set_2 is superset of set_1

We have defined two sets referenced by variables set_1 and set_2. Both sets are empty sets. Using built-in issuperset() method we are checking if set_1 is superset of set_2 OR set_2 is superset of set_1. Output from issuperset() method will be a boolean value.

Program Output

using issuperset() method - with empty set
set()         # elements of set_1
set()         # elements of set_2
True         # set_1 is superset of set_2
True         # set_2 is superset of set_1

Since both set_1 and set_2 are empty sets, so they are superset to each other. As an output we got True for both set_1 superset of set_2 and set_2 superset of set_1.

Using superset (>=) operator

• Syntax – set_1 >= set_2
• set_1 >= set_2 returns boolean (True or False) value. True means set_1 is a superset of set_2 and False means set_1 is not a superset of set_2.

# using superset (>=) operator
print("using superset (>=) operator")
set_1 = {10, 20, 30}
set_2 = {10, 20, 30, 70, 80, 90}
superset_set12 = set_1 >= set_2            # check set_1 is superset of set_2
superset_set21 = set_2 >= set_1            # check set_2 is superset of set_1
print(set_1)
print(set_2)
print(superset_set12)       # will return False = set_1 is not superset of set_2
print(superset_set21)       # will return True = set_2 is superset of set_1

We have defined two sets referenced by variables set_1 and set_2. Using superset (>=) operator, we are checking if set_1 and set_2 are superset to each other or not. Output from superset >=) operator will be a boolean value.

Program Output

using superset (>=) operator
{10, 20, 30}                          # elements of set_1
{80, 20, 90, 70, 10, 30}       # elements of set_2
False              # set_1 is not superset of set_2
True               # set_2 is superset of set_1

From the program output, set_1 and set_2 elements are printed. superset (>=) operator has return a boolean (True or False) value. False shows that set_1 is not superset of set_2 and True shows that set_2 is superset of set_1.

Using proper (strict) superset (>) operator

• Syntax – set_1 > set_2 (Note – we have only greater than operator, equal to sign is not with it)
• set_1 > set_2 returns boolean (True or False) value. True means set_1 is a superset of set_2 and False means set_1 is not a superset of set_2.
• This operator is useful to check if set_1 is proper superset of set_2 or not.

# using strict superset (>) operator
print("using strict superset (>) operator")
set_1 = {10, 20, 30}
set_2 = {10, 20, 30, 70, 80, 90}

superset_set12 = set_1 > set_2          # check set_1 is proper superset of set_2
superset_set21 = set_2 > set_1          # check set_2 is proper superset of set_1

print(set_1)
print(set_2)
print(superset_set12)      # will return False = set_1 is not superset of set_2
print(superset_set21)      # will return True = set_2 is superset of set_1

We have defined two sets referenced by variables set_1 and set_2. Using proper (strict) superset (>) operator, we are checking if set_1 and set_2 are proper superset to each other or not. Output from proper (strict) superset (>) operator will be a boolean value.

Program Output

using strict superset (>) operator
{10, 20, 30}                        # elements of set_1
{80, 20, 90, 70, 10, 30}     # elements of set_2
False             # set_1 is not superset of set_2
True              # set_2 is superset of set_1

From the program output, set_1 and set_2 elements are printed. Proper (strict) superset (>) operator has return a boolean (True or False) value. False shows that set_1 is not proper superset of set_2 and True shows that set_2 is proper superset of set_1.

Custom implementation – issuperset() method

Internally in Python, issuperset() method checks for all elements of set_2 are present in set_1. If any element of set_2 is not present in set_1, then method returns False; otherwise, it returns True.

Lets see the custom code to implement functioning of issuperset() method.

# Identifies if set_1 is superset of set_2
def custom_issuperset(set_1, set_2):
    # Iterate through elements of set_2
    for element in set_2:
        # Check if any element of set_2 not in set_1
        if element not in set_1:
            return False
    return True

set_1 = {10, 20, 30}
set_2 = {10, 20, 30, 40, 50, 60}
set_3 = set()
set_4 = set()
print(custom_issuperset(set_1, set_2))  # Output: False
print(custom_issuperset(set_2, set_1))  # Output: True
print(custom_issuperset(set_3, set_4))  # Output: True

• custom_issuperset() function iterates through each element in set_2.
• For each element in set_2, it checks if the element is present in set_1 or not.
• If any element is not found in set_1, the function returns False, else it will continue to check for all the elements of set_2. If all elements of set_2 are present in set_1, then it will return True.

Summary

In this article we learn about various subset operation on Sets. Following scenarios were explored:

• Superset – Mathematical Definition
• Using issuperset() method
• Using issuperset() method – with empty sets
• Using superset (>=) operator
• Using proper (strict) superset (>) operator
• Custom implementation – issuperset() method

Code – Github Repository

All code snippets and programs for this article and for Python tutorial, can be accessed from Github repository – Comments and Docstring in Python.

Interview Questions & Answers

Q: Can issuperset() be used with other iterables like lists and dictionary?

Yes, the issuperset() method can be used with any iterable e.g., lists, tuples, etc. The method checks whether all elements of the iterable are present in the set from which issuperset() is called.

• With lists, it checks if all elements of the list are present in the set.
• With dictionaries it checks if all dictionary keys are present in the set.

set_1 = {10, 20, 30, 40}
list_1 = [10, 20, 30]

print(set_1.issuperset(list_1))       # True

dict_1 = {10: 'Python', 20: 'Power BI', 30: 'LLM'}
print(set_1.issuperset(dict_1))      # True

Q: What is the time complexity of the issuperset() method?

Time complexity of the issuperset() method is O(n), where n is the number of elements in the set provided as an argument. The method checks for the presence of each element of the provided set in the original set.

Q: Can issuperset() be used to compare sets with different data types?

Yes, issuperset() can be used to compare sets that contain different data types. The method checks for the presence of elements, irrespective of their data types.

Q: What happens if the argument passed to issuperset() is not iterable?

If a non-iterable object (like an integer) is passed to the issuperset() method, a TypeError will be raised. This is because the method expects an iterable to check the elements against the set.

set_1 = {10, 20, 30, 40}
try:
    print(set_1.issuperset(10))
except TypeError as e:
    print(e)     # 'int' object is not iterable