DISJOINT UNION Meaning and
Definition
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The term "disjoint union" refers to a mathematical concept that combines two or more sets into a new set such that no elements are shared between them. It is also known as a "tagged union" or a "coproduct." When sets are disjoint, it means they have no common elements or overlap.
In the context of set theory, the disjoint union of two sets A and B is denoted as A ⊔ B. It consists of all elements from A and B, but each element is uniquely tagged to indicate which set it belongs to. For example, if A = {1, 2} and B = {3, 4}, then A ⊔ B would be {1A, 2A, 3B, 4B}.
The disjoint union operation constructs a larger set from smaller sets, preserving their individual characteristics. It is commonly used in various branches of mathematics, such as algebra, topology, and category theory. In algebraic structures like groups and rings, the disjoint union serves as a way to combine objects with different properties and define operations on them.
The concept of a disjoint union can also be applied beyond sets. For instance, in computer science and programming languages, disjoint unions (also called sum types) are used to define data types that can have multiple alternative forms. Each form is associated with a unique tag, which helps determine the current form of the data and allows for pattern matching and type checking.
Etymology of DISJOINT UNION
The word "disjoint" comes from the Old French verb "desjoindre", which is a combination of the prefix "des-" (meaning "apart" or "undo") and the verb "joindre" (meaning "join" or "unite"). "Disjoint" refers to the process of separating or disconnecting.
The term "union" comes from the Latin word "unio", which means "unity" or "joining together".
So, when combined, "disjoint union" refers to a union or joining together, but in a way that separates or disconnects the elements being united. It is commonly used in mathematics to describe the construction or combination of two or more disjoint sets or groups.
