ekvivalenssiluokassaryhmä

ekvivalenssilu is a Finnish term that can be translated as "equivalence class" or "class of equivalence." In mathematics, an equivalence class is a subset of the elements of a set that are related to each other by a given equivalence relation. An equivalence relation is a binary relation that is reflexive, symmetric, and transitive.

For example, consider the set of all integers and the relation "has the same remainder when divided by 3." This is an equivalence relation. The integers can be partitioned into three equivalence classes: those with a remainder of 0 when divided by 3 (..., -3, 0, 3, 6, ...), those with a remainder of 1 (..., -2, 1, 4, 7, ...), and those with a remainder of 2 (..., -1, 2, 5, 8, ...). Each of these sets is an equivalence class.

The concept of equivalence classes is fundamental in many areas of mathematics, including abstract algebra, topology, and set theory. It allows mathematicians to group elements that share a common property and to study the properties of these groups rather than individual elements. This simplification can lead to deeper insights and more elegant proofs. The set of all equivalence classes of a set with respect to an equivalence relation is called the quotient set.