Why did the man choose to live and mother did not?

Mother understands the world is dead and refuses to starve or fall victim to savage marauders. She wanted her boy to die as well so he can escape a horrid fate. Father's only interest is the boy. He sees goodness and hope in the boy even though there is no hope for the world.

The child is my warrant, and if he is not the word of God, then God never spoke.

The man is not sure what good it will do keeping the boy alive: he just knows he has to try. The man exists only for the boy's sake.