"Do not think that I came to bring peace on earth. I did not come to bring peace but a sword. For I have come to set a man against his father, a daughter against her mother, and a daughter-in-law against her mother-in-law; and a 'a man's enemies will be those of his own household.' He who loves father or mother more than Me is not worthy of Me. And he who loves son or daughter more than Me is not worthy of Me. And he who does not take his cross and follow after Me is not worthy of Me. He who finds his life will lose it, and he who loses his life for My sake will find it."
Although Jesus did not come to purposefully cause the disturbances described here, such is a natural consequence of His teachings. That which is good and pure will be at variance with that which is evil and defiled. Truth will battle error because holiness hates sin (cf. Psa. 119:104). Even though Jesus' ministry did not bring earthly peace, it did make true spiritual peace possible both now and in the afterlife for His faithful followers (cf. Phil. 4:7).
Jesus' description in Matthew 10:35,36 is not what one would expect. Generally, immediate family members share strong relationships. Fathers and sons and mothers and daughters typically enjoy solid bonds, especially after the children have matured to adulthood. However, the gospel is a sword that brings division even to the strongest of family ties. When one believes in the Lord Jesus, obeys the gospel, and dedicates his life to righteousness and the spread of the truth, there will inevitably be conflict with those who reject Christ--even if they are beloved flesh-and-blood relatives. In fact, Jesus here teaches that one may become enemies with his family members because of the gospel (cf. Micah 7:6)!
It is certainly the case that no child of God would labor to make an enemy of anyone--especially not an immediate family member. Nevertheless, such is sometimes unavoidable for the Christian who is forced to make choices between allegiance to relatives or the Lord and between pleasing family or Almighty God. Jesus addressed this point in Matthew 10:37 - "He who loves father or mother more than Me is not worthy of Me." Even the closest of family ties must be disregarded when they come into conflict with one's duty to God. Jesus must be second to none so far as our priorities are concerned! We must love Him above all others, otherwise we are unworthy of Him. It should be remembered that truth is more authoritative than a parent is and that which Jesus has done for us far exceeds anything our nearest relative can do. We should not even allow our beloved children to hinder our walk with the Lord.
It is sad that even today the love for false religion is strong in many families. For example, those who leave Islam today for Christianity may be forsaken by their families. Now, since these families value their false religion to such a degree, how much more should a Christian value the only true religion as more precious than his family ties?
Our Lord plainly affirmed in 10:38 - "He who does not take his cross and follow after Me is not worthy of Me" (cf. Luke 9:23; Rom. 12:1; Gal. 2:20). When a criminal was condemned to crucifixion, part of his sentence was that he should carry his cross to the place of execution. For one to take up a cross and follow after Jesus would imply that Jesus Himself would carry a cross. Although this truth is not profound from our perspective, this is definitely an allusion to the manner of Jesus' death (a fact which His disciples didn't fully understand at the time). He suffered according to the full-measure of divine displeasure for the sins of the world. Therefore, all should be willing to cheerfully follow Him, even if it results in the light cross of persecution and division within one's family. To fail to take up this cross is to show one's ingratitude toward Jesus and the salvation made possible through His crucifixion. Such a one is unworthy of Christ and His saving gospel!
"He who finds his life will lose it, and he who loses his life for My sake will find it" (Matt. 10:39). He who preserves his physical life at the expense of duty and faithfulness shall lose eternal life. Although one who neglects to bear his cross will feel that he has found a more enjoyable physical life because he has escaped the suffering of persecution and the pain of family discord, in actuality he has forfeited true life. In contrast, he who sacrifices his physical life in faithful service to God will inherit everlasting life.