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The idea of sets is crucial in mathematics. In the late 1800s, the present discipline of set theory was codified. Set theory is a fundamental language of mathematics and the repository of modern mathematics’ fundamental notions. On the other hand, it is a discipline of mathematics in its own right, classed in modern mathematics as a branch of mathematical logic.
A set is a well-defined grouping of items. Well-defined indicates that there is a system in place to identify whether a given item belongs to a specific set or not. Elements or members of a set are objects that belong to it. Capital letters are used to indicate sets, whereas lower case letters are used to represent components.
Subset and Superset
Subset
Because a set is a well – defined collection of objects or items organised together inside braces, it may be divided into smaller sets known as subsets. A set A is said to be the subset of another set B if every member of set A is also an element of set B. Exemplifications –
Consider a set X such that X = Set of all the people residing in your city, and another set Y = Set of your family members is a subset of X since each member of your family is also a resident of the city in which you live. Because every member of set Y is a component of several elements of set X, Y is unquestionably a subset of set X.
If set E is defined as the set of all even numbers and set N is defined as the set of all natural numbers, then set E is a subset of set N.
Superset
Supersets are those sets that are specified by the requirements A B and A B. When these two requirements are met, B is referred to as a superset of set A.
Examples: A = Set of all polygons and B = Set of regular polygons; hence, A is the superset of set B in this B A and B A.
X ={ 1, 2, 3, 4, 5, 6 }and Y = {s: s< 4 and sϵ N}; in this situation, set Y is a subset of set X and set X is a superset of set Y.
Superset-Related Properties
Every set is a superset of a null, void, or empty set, i.e. A, since it has no items.
Because each set is a subset of itself, each set is also a superset of itself;
Properties of Subsets and Supersets
Properties of Superset
After you’ve mastered the definition, notation, and comparison with subsets, it’s time to move on to the topic’s crucial properties:
• Any set is a superset of an empty set; this is because a null/empty set does not contain any elements.
This may be expressed symbolically as: For each set X, we can say that X= ∅.
• If LCM then ML is true for the supplied two sets (L and M). This means that the superset and subset are diametrically opposed.
• A superset from the provided get is the one with the most items (all of them); on the other hand, a subset has fewer elements.
Differences of subsets and supersets
If and only if every element of set A is also an element of set B, then set A is said to be a subset of set B. A B denotes such a relationship between sets. It is also possible to interpret it as ‘A is included in B.’ If A B and A B, and indicated by A B, the set A is said to be a proper subset. If A has even one person who is not a member of B, A cannot be a subset of B. An empty set is a subset of any set, therefore a set is a subset of itself.
A is contained in B if it is a subset of B. It indicates that B contains A, or that B is a subset of A. To indicate that B is a superset of A, we write A B.
A ={ 1, 3} is a subset of B ={ 1, 2, 3} because all of the components in A are contained in B. Because B contains A, it is a superset of A. Assume A={1, 2, 3} and B={ 3, 4, 5}. Then AB={ 3}. As a result, A and B are both supersets of AB. Because AB contains all of the items in A and B, it is a superset of both A and B.
If A is a subset of B and B is a subset of C, then A is also a subset of C. Any set A is a superset of the empty set, and any other set is a superset of that set.
Conclusion
A superset is a set of another smaller set in which practically all of the smaller set’s components are also members of the set. We know that if B is included inside A, then A contains B. To put it another way, if B is a subset of A, then A is its superset.
