Basic Properties of Even and Odd Functions

In this paper x, r are real numbers. Let A be a set. We say that A is symmetrical if and only if: (Def. 1) For every complex number x such that x ∈ A holds −x ∈ A. Let us note that there exists a subset of C which is symmetrical. Let us observe that there exists a subset of R which is symmetrical. In the sequel A denotes a symmetrical subset of C. Let R be a binary relation. We say that R has symmetrical domain if and only if: (Def. 2) domR is symmetrical. Let us observe that every binary relation which is empty has also symmetrical domain and there exists a binary relation which has symmetrical domain. Let R be a binary relation with symmetrical domain. Observe that domR is symmetrical. Let X, Y be complex-membered sets and let F be a partial function from X to Y . We say that F is quasi even if and only if: