Mat 10:35 For I have come to set a man at variance against his father, and the daughter against her mother, and the daughter in law against her mother in law.

"To set at variances" is the Greek dichazo, which means "to divide in two." It only appears once in the Gospels, here. The root word, schazo, means "to cut." It is related, but not the same word as dia-schizô, which means "to cleave asunder" and "set at variance" and diakeazô which also means "to cleave asunder."

"Against" is kata, which means primarily "downwards," but in various usages comes to mean "through out," "according to," "along," "against," and "toward." It is most often translated in the KJV as "according to" and "after." It has this sense of downward motion and can mean "descended from" or "divided from." It is often used in compound Greek words.

The Greek term kata-schazô combines the two key words of this verse and means "cut into."

So Christ isn't saying the he has come to set people in opposition to one another, but that he is cutting them off from one another. He is starting something new, something descended from what exists now, but apart from it.