- Kernel (set theory)
mathematics, the kernel of a function "f" may be taken to be either
equivalence relationon the function's domain that roughly expresses the idea of "equivalent as far as the function "f" can tell", or
*the corresponding partition of the domain.
Note that there are several other meanings of the word "kernel" in mathematics; see
kernel (mathematics)for these.
For the formal definition, let "X" and "Y" be sets and let "f" be a function from "X" to "Y".Elements "x"1 and "x"2 of "X" are "equivalent" if "f"("x"1) and "f"("x"2) are equal, i.e. are the same element of "Y".The kernel of "f" is the equivalence relation thus defined.
The kernel, in the equivalence-relation sense, may be denoted "="f"" (or a variation) and may be defined symbolically as
equivalence relation, the kernel can be modded out to form a quotient set, and the quotient set is the partition:
This quotient set "X"/="f" is called the "
coimage" of the function "f", and denoted "coim "f" (or a variation).The coimage is naturally isomorphic (in the set-theoretic sense of a bijection) to the image, im "f"; specifically, the equivalence classof "x" in "X" (which is an element of coim "f") corresponds to "f"("x") in "Y" (which is an element of im "f").
binary relation, the kernel of a function may be thought of as a subsetof the Cartesian product"X" × "X".In this guise, the kernel may be denoted "ker "f" (or a variation) and may be defined symbolically as
But this is not useful merely as a formalisation in
set theory!In fact, the study of the properties of this subset can shed important light on the function in question.We give here two examples.
First, if "X" and "Y" are
algebraic structures of some fixed type (such as groups, rings, or vector spaces), and if the function "f" from "X" to "Y" is a homomorphism, then ker "f" will be a subalgebraof the direct product"X" × "X". Subalgebras of "X" × "X" that are also equivalence relations (called " congruence relations") are important in abstract algebra, because they define the most general notion of quotient algebra. Thus the coimage of "f" is a quotient algebra of "X" much as the image of "f" is a subalgebraof "Y"; and the bijection between them becomes an isomorphismin the algebraic sense as well (this is the most general form of the first isomorphism theoremin algebra). The use of kernels in this context is discussed further in the article Kernel (algebra).
Secondly, if "X" and "Y" are
topological spaces and "f" is a continuous functionbetween them, then the topological properties of ker "f" can shed light on the spaces "X" and "Y".For example, if "Y" is a Hausdorff space, then ker "f" must be a closed set.Conversely, if "X" is a Hausdorff space and ker "f" is a closed set, then the coimage of "f", if given the quotient space topology, must also be a Hausdorff space.
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