- Kernel (set theory)
In

mathematics , the**kernel**of a function "f" may be taken to be either*the

equivalence relation on 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: $(x\; =\_\{f\}\; y);\; :Longleftrightarrow;\; (f(x)\; =\; f(y)).\; !$

Like any

equivalence relation , the kernel can be modded out to form aquotient set , and the quotient set is the partition::$left\{,\; left\{,\; w\; in\; X:\; f(x)=f(w)\; ,\; ight\}\; :\; x\; in\; X\; ,\; ight\}.$

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 abijection ) to the image, im "f"; specifically, theequivalence class of "x" in "X" (which is an element of coim "f") corresponds to "f"("x") in "Y" (which is an element of im "f").Like any

binary relation , the kernel of a function may be thought of as asubset of theCartesian product "X" × "X".In this guise, the kernel may be denoted "ker "f" (or a variation) and may be defined symbolically as: $mathop\{mathrm\{ker\; f\; :=\; \{(x,x\text{'})\; mid\; f(x)\; =\; f(x\text{'})\}mbox\{.\}\; !$

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 structure s of some fixed type (such as groups, rings, orvector space s), and if the function "f" from "X" to "Y" is ahomomorphism , then ker "f" will be asubalgebra of thedirect product "X" × "X". Subalgebras of "X" × "X" that are also equivalence relations (called "congruence relation s") are important inabstract algebra , because they define the most general notion ofquotient algebra . Thus the coimage of "f" is a quotient algebra of "X" much as the image of "f" is asubalgebra of "Y"; and the bijection between them becomes anisomorphism in the algebraic sense as well (this is the most general form of thefirst isomorphism theorem in algebra). The use of kernels in this context is discussed further in the articleKernel (algebra) .Secondly, if "X" and "Y" are

topological space s and "f" is acontinuous function between them, then the topological properties of ker "f" can shed light on the spaces "X" and "Y".For example, if "Y" is aHausdorff space , then ker "f" must be aclosed 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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