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In mathematics, what is a subset? Consider the following real-world example: Assume you have hundreds of music on your iPod. If someone just wants to listen to rock tunes, they can make a playlist on their iPod that exclusively contains rock songs. This playlist is a subset of the music already on the iPod.
A subset is any group of items that are part of a larger set, such as a playlist of music drawn from a bigger list of songs. The notion of this subset in mathematics is useful and applies to many different fields of mathematics. So, what exactly is a subset? The formal definition of a subset follows.
Subset Definition
What does the term “subset” mean? A set is a collection of items that can be elements, numbers, variables, symbols, or anything else. This leads to the following subset definition: Set A is a subset of set B if all of its items are also elements of B. Set A is a subset of the set if it is totally contained within the other set.
Compare this definition to the iPod as an example. Set A is a playlist of rock music, while Set B contains the totality of the songs on the iPod. Because set A contains all of the songs in set B, A is a subset of B.
Because set A is contained within set B, it is a subset of B. Subsets are represented by the notation AB. An appropriate subset is one that is not the same as the overlaying set. AB is the suitable subset symbol used throughout mathematics. In this scenario, the subset may be equal to the underlying set.
Properties of Subsets
Subsets are one of the most fundamental ideas in set theory. To emphasise the significance of subsets, it is vital to analyse their qualities. The following are some of the most important subset properties.
It Is a Subset of Itself
Every set is thought to be a subset of itself.
Whether we have a limited or infinite set, a set is considered a subset of itself. This occurs without exception. We shall always include the set itself as a subset when listing the subsets of any given set.
In the case of appropriate subsets, however, we shall skip the set to make the subset equal to the set.
For a finite set A = 2,5, for example, all potential subsets are:
A = Ⲫ, A = {2}, A = {5}, A = {5}, A = {2, 5}
To meet the property, we have included a subset with the same items as the original set.
As previously stated, this is not limited to finite sets; infinite sets exhibit the same feature.
Subset of an Empty Set
Every set’s subset is the empty set.
Consider set A, which might be finite or infinite. We can compute all of A’s potential subsets; among these subsets, we will include a null/empty set.
Consider the finite set A = 1, 2; all the potential subsets of this set are:
A = Ⲫ, A = {1}, A = {2}, A = {1, 2}
As you can see, we added an empty subset in our list of subsets to meet the property:
Ⲫ ⊂ A
The same concept may be applied to infinite sets. It makes no difference if a set is finite or infinite; an empty set is always a subset of the provided set.
The Intersection of Two Sets
Set A is a subset of set B if and only if the intersection of A and B is equal to A.
If a given finite or infinite set A is a subset of any finite/infinite set B, their intersection must always equal set A. This is one of the prerequisites for set A being a subset of set B. If this criterion is not met, we may readily conclude that set A is not a subset of set B.
This may be written as:
A ⊂ B 🡪 A ∩ B = A
Union of Two Sets
Set A must be a subset of set B if their union is equal to set B.
To be a subset of any finite or infinite set B, a given finite or infinite set A’s union must always be identical to set B. This is one of the prerequisites for set A being a subset of set B. If this criterion is not met, we may readily conclude that set A is not a subset of set B.
This may be written as:
A ⊂ B 🡪 A ∪ B = B
Symbols in Subsets
In set theory, a subset is represented by the sign cand, which can be translated as ‘is a subset of’ in many languages.
Subsets can be represented in the following ways using this symbol:
P⊆Q, which may be interpreted as Set P is a subset of Set Q, is a subset of Set Q.
Take note that a subset can be identical to the set, that is, a subset can contain all of the items that are contained inside the set.
Conclusion
One is the significance of sets. They allow us to handle a collection of mathematical objects as if they were a separate mathematical entity. We can worm our way around sets when working with finite collections of items.
