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Prove or disprove that f is a quotient mapping.
f:R^3\{(x1,x2,x3):x1=0}--->R^2 defined by
(x1,x2,x3)|->(x2/x1,x3/x1)
f:R^3\{(x1,x2,x3):x1=0}--->R^2 defined by
(x1,x2,x3)|->(x2/x1,x3/x1)
A mapping, also known as a function, is a mathematical concept that describes how elements from one set are related to elements in another set. It is denoted by f(x) and is used to transform input values into output values.
To verify if a mapping is quotient, you need to check if it satisfies the properties of a quotient mapping. These properties include being surjective (onto), preserving equivalence classes, and being continuous. If the mapping satisfies all of these properties, it is a quotient mapping.
Verifying if a mapping is quotient is important because it ensures that the mapping is well-defined and that the equivalence classes are preserved. This is necessary for making accurate mathematical statements and proofs.
No, a mapping cannot be both quotient and injective. A quotient mapping is surjective (onto), while an injective mapping is bijective (one-to-one). Therefore, a mapping cannot have both properties simultaneously.
To prove that a mapping is quotient, you need to show that it satisfies the three properties of a quotient mapping: surjectivity, preservation of equivalence classes, and continuity. This can be done by using mathematical definitions and principles, and providing logical reasoning and evidence.