In a message dated 98-09-17 16:03:39 EDT, Mary Harrison wrote:
<< Topic of conversation in the math office today: Is it possible for a function or relation to be both symmetric to the y axis and to the origin but not to the x-axis?
The only thing I could think of was 3/4 of a circle.
Anything else? Other than variations of this (e.g. triangle segments) >> 3/4 circle is not symmetric to the origin!! The answer to the question is "NO." If the relation R is symmetric wrt y axis and origin, and (x, y) is an element of R, then (-x, y) is an element of R as is (-x, -y). Then, since (-x, y) is an element of R, (x, -y) -- the reflection of (-x, y) wrt the origin -- is also an element of R. So (x, y) element of R implies (x, -y) is an element of R, and R is also symmetric wrt x axis!!