A subring I of a ring is called a left ideal if given any element x of the subring I the product yx is in I for any choice of y from the original ring. An analogous definition is made for right ideals with respect to multiplication on the right. If an ideal is both a left and right ideal it is called a two-sided ideal or simply an ideal. The kernel of a (ring) homomorphism provides the canonical example of an ideal.
A technique for counting the elements of a set that don't have certain properties. Let S be a set of n-elements, and let P1 and P2 be properties that the elements of S may or may not have. Let A1 be the subset of S consisting of those elements having P1 and A2 be those having P2 . The number of elements of S having neither P1 nor P2 is given by:
|S|-|A1|-|A2|+|A1 intersect A2|
The last term has to be added because the elements having both properties have been subtracted twice in the previous terms.
This is the general idea of the inclusion/exclusion principle, the formula becomes slightly more complicated but can readily be generalized to n properties.
In a directed graph, we say that a vertex has indegree x if there are (exactly) x edges coming into that vertex.
A set of vertices in a graph none of which are connected by an edge.
An element of order 2 in a group.