Definition. A graph is an ordered pair $G=(V,E)$ consisting of a set of vertices $V=\{v_1,v_2,\dots,v_n\}$ and edges $E\subseteq V\times V.$
Definition. A matching is a subset $M\subseteq E$ such that no two edges in $M$ are adjacent, i.e., $$ \forall j \neq k\in V,\quad (i,j) \in M \implies (i,k) \notin M. $$
Definition. A matching $M$ is called maximal if $M$ cannot be extended by adding any more edges, i.e., adding any extra edge will break the matching.
Definition. A matching $M$ is called maximum if $M$ has the largest possible number of edges.
Definition. A matching $M$ is called perfect if $M$ matches all the vertices.
Theorem. The symmetric difference of two matchings is a graph in which every component is either a path or an even cycle.
Proof. Every vertex in $M_1\bigtriangleup M_2$ has degree at most $2.$ Therefore the components are either paths or cycles.
In any cycle $C\in M_1\bigtriangleup M_2,$ every vertex has exactly one $M_1$ edge and one $M_2$ edge.
In any cycle, number of $M_1$ edges is equal to number of $M_2$ edges. Hence even. $\blacksquare$
