For the formal definition, let *X* and *Y* be sets and let *f* be a function from *X* to *Y*.
If *x* and *x*' are elements of *X*, then *x* and *x*' are *equivalent* if *f*(*x*) and *f*(*x*') are equal as elements of *Y*.
The kernel of *f* is the equivalence relation thus defined.

The kernel may be denoted "=_{f}" (or a variation) and may be defined symbolically as

Like any binary relation, the kernel of a function may be thought of as a subset of the Cartesian product *X* × *X*.
In this guise, the kernel may be denoted "ker *f*" (or a variation) and may be defined symbolically as

First, if *X* and *Y* are algebraic structures of some fixed type (such as groupss, ringss, or vector spaces), and if the function *f* from *X* to *Y* is a homomorphism, then ker *f* will be a subalgebra of 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 subalgebra of *Y*; and the bijection between them becomes an isomorphism in the algebraic sense as well (this is the most general form of the first isomorphism theorem in 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 function between 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 Hausdorff and ker *f* is a closed set, then the coimage of *f*, if given the quotient space topology, must also be Hausdorff.