Kernel (set theory)

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In set theory, the kernel of a function ${\displaystyle f}$ (or equivalence kernel^[1]) 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 ${\displaystyle f}$ can tell",^[2] or
• the corresponding partition of the domain.

An unrelated notion is that of the kernel of a non-empty family of sets ${\displaystyle {\mathcal {B}},}$ which by definition is the intersection of all its elements: $${\displaystyle \ker {\mathcal {B}}~=~\bigcap _{B\in {\mathcal {B}}}\,B.}$$ This definition is used in the theory of filters to classify them as being free or principal.