Non-Jew at Seder

Question

You stated that it is absolutely permitted to have a non-Jew at the seder. Why isn't this a problem of "inviting a non-Jew for Yom Tov", prohibited lest you cook for them, etc. CYA asked one correspondent where it is brought down that "becoming aware" of a mixture is a precondition of bittul. This concept is explicitly mentioned in Yoreh Deah where the issue of "Chozer v'neyor" is discussed - with regard to the case of putting a meat spoon into a milk pot twice.