What is equivalence class? Explain with example and vedio.
What is equivalence class? Explain with example and vedio.
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements aand b belong to the same equivalence class if and only if a and b are equivalent.
Formally, given a set S and an equivalence relation ~ on S, the equivalence class of an element a in S is the set
{\displaystyle \{x\in S\mid x\sim a\}}
of elements which are equivalent to a. It may be proven from the defining properties of "equivalence relations" that the equivalence classes form a partition of S. This partition – the set of equivalence classes – is sometimes called the quotient set or the quotient space of S by ~ and is denoted by S / ~.
When the set S has some structure (such as a group operation or a topology) and the equivalence relation ~ is defined in a manner suitably compatible with this structure, then the quotient set often inherits a similar structure from its parent set. Examples include quotient spaces in linear algebra, quotient spaces in topology,
- If X is the set of all cars, and ~ is the equivalence relation "has the same color as", then one particular equivalence class consists of all green cars. X/~ could be naturally identified with the set of all car colors.
- Let X be the set of all rectangles in a plane, and ~ the equivalence relation "has the same area as". For each positive real number A there will be an equivalence class of all the rectangles that have area A.[1]
- Consider the modulo 2 equivalence relation on the set Z of integers: x ~ y if and only if their difference x − y is an even number. This relation gives rise to exactly two equivalence classes: one class consisting of all even numbers, and the other consisting of all odd numbers. Under this relation [7], [9], and [1] all represent the same element of Z/~.[2]
- Let X be the set of ordered pairs of integers (a,b) with b not zero, and define an equivalence relation ~ on X according to which (a,b) ~ (c,d) if and only if ad = bc. Then the equivalence class of the pair (a,b)can be identified with the rational numbera/b, and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of rational numbers.[3] The same construction can be generalized to the field of fractions of any integral domain.
- If X consists of all the lines in, say the Euclidean plane, and L ~ M means that Land M are parallel lines, then the set of lines that are parallel to each other form an equivalence class as long as a line is considered parallel to itself. In this situation, each equivalence class determines a point at infinity.
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements aand b belong to the same equivalence class if and only if a and b are equivalent.
Formally, given a set S and an equivalence relation ~ on S, the equivalence class of an element a in S is the set
{\displaystyle \{x\in S\mid x\sim a\}}of elements which are equivalent to a. It may be proven from the defining properties of "equivalence relations" that the equivalence classes form a partition of S. This partition – the set of equivalence classes – is sometimes called the quotient set or the quotient space of S by ~ and is denoted by S / ~.
When the set S has some structure (such as a group operation or a topology) and the equivalence relation ~ is defined in a manner suitably compatible with this structure, then the quotient set often inherits a similar structure from its parent set. Examples include quotient spaces in linear algebra, quotient spaces in topology,
- If X is the set of all cars, and ~ is the equivalence relation "has the same color as", then one particular equivalence class consists of all green cars. X/~ could be naturally identified with the set of all car colors.
- Let X be the set of all rectangles in a plane, and ~ the equivalence relation "has the same area as". For each positive real number A there will be an equivalence class of all the rectangles that have area A.[1]
- Consider the modulo 2 equivalence relation on the set Z of integers: x ~ y if and only if their difference x − y is an even number. This relation gives rise to exactly two equivalence classes: one class consisting of all even numbers, and the other consisting of all odd numbers. Under this relation [7], [9], and [1] all represent the same element of Z/~.[2]
- Let X be the set of ordered pairs of integers (a,b) with b not zero, and define an equivalence relation ~ on X according to which (a,b) ~ (c,d) if and only if ad = bc. Then the equivalence class of the pair (a,b)can be identified with the rational numbera/b, and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of rational numbers.[3] The same construction can be generalized to the field of fractions of any integral domain.
- If X consists of all the lines in, say the Euclidean plane, and L ~ M means that Land M are parallel lines, then the set of lines that are parallel to each other form an equivalence class as long as a line is considered parallel to itself. In this situation, each equivalence class determines a point at infinity.
