In Ayn Rand's Anthem, who is Liberty 5-3000 and why does Equality like her?

When Equality first sees Liberty, he sees that her "body was straight and thin as a blade of iron," and her "eyes were dark and hard and glowing, with no fear in them, no kindness and no guilt." Her hair is "golden as the sun" and it flies in the wind, "shining and wild, as if it defied men to restrain it." He associates her with a metal that is incredibly strong—iron—and he takes note of the fact that there is no fear in her eyes. Further, he describes her hair, and by extension, herself, as wild and incapable of being restrained. Her appearance seems to connote an intelligence and courageousness that Equality shares. She works the soil and lives in the homes of the peasants outside of the city, and Equality sees her each day when the street sweepers clean the road to the first milepost outside the city. Equality seems to be attracted to her strength and her courage. She is unafraid to acknowledge him, and her hair and her eyes seem to betoken her bravery and fortitude.

Liberty is introduced in chapter two after Equality has described most of the workings and daily life of the society in which he lives. Equality is attracted to Liberty because she is different than the other women with whom she works in the fields. Not only is she a beautiful blond, but her body language suggests to him that she is different. She is strong and confident, for example, which he himself is as well. He recognizes immediately that she is one who would not walk in fear, which he also analyzes as he compares some other brothers who live at his home with him. He describes them as hunched over and never looking at anyone in the eyes; but Liberty greeted him every day "with [her] eyes" (40) which suggests that she knows no fear and is willing to break rules to meet her own desires like he does.

How does Equality 7-2521's love for Liberty 5-3000, desire to "raze the cities of the enslaved," and passion for science in Anthem demonstrate his unselfishness?

In Ayn Rand's Anthem, Equality 7-2521 lives in a society which has attempted to abolish the ego, and selfishness along with it. Throughout the narrative, Equality learns to assert himself against this tyrannical system, finding that his own desires are a reliable guide. However, his motivations are still not primarily selfish.

Equality's love for Liberty 5-3000 is depicted as a strong natural emotion, which is fully reciprocated, in the midst of an atmosphere of coercion, and which shows that the state has not succeeded in crushing Equality's spirit. The way in which Equality behaves to Liberty shows his respect for her, and the unselfish nature of his feelings. She is always a free agent, telling him how she feels, following him independently to the forest, and vowing her devotion to him.

An even clearer indication of Equality's unselfish nature is his final vow to fight for those who remain enslaved. At the end of the book, he promises:

And the day will come when I shall break all the chains of the earth, and raze the cities of the enslaved, and my home will become the capital of a world where each man will be free to exist for his own sake. For the coming of that day shall I fight, I and my sons and my chosen friends. For the freedom of Man. For his rights. For his life. For his honor.

There is clearly an irony here. Equality vows to fight for "the sacred word: EGO." However, his struggle on behalf of the egos of others is altruistic.

Finally, Equality's passion for science and his pursuit of knowledge are shown as disinterested, since they bring him no benefit, only punishment and banishment. It is true that he initially thinks he will be rewarded for his work, but he remains dedicated after it becomes clear that this is not the case.

In Ayn Rand's Anthem, how is Equality 7-2521's love for Liberty 5-3000 meaningful and unselfish?

While the kind of romantic love Equality 7-2521 feels for Liberty 5-3000 has elements of selfishness, as romantic love always does, it also has strong elements of unselfishness. Interestingly, Ayn Rand is famous for arguing that selfishness is a virtue and that the highest good comes from pursuing one's own self interest.

Equality's love for Liberty is meaningful to him not simply because it benefits himself, but because he can benefit her for her own sake by setting her free. He realizes, after he leaves the City and finds the house in the Uncharted Forest, that collectivism is a great evil. Although Liberty 5-3000, who has a left the City to follow him, wants only to obey him, he does not want that kind of mindless devotion from her. He wants her to be her own person, even if it makes his life harder, because he has had his revelation that:

And I shall choose only such as please me, and them I shall love and respect, but neither command nor obey.

It might be easier for Liberty, now renamed Gaea, to obey as she has been trained all her life to do, and easier for Equality, now named Prometheus, to command her, but he puts the principle of individual freedom first.

Prometheus thus shows unselfish love towards Gaea when he allows her to do what she wants. For example, he leaves her to look at herself in the mirror they have discovered while he does heavy labor of gathering wood, hauling water, and killing a mountain goat to cook. He mentions that "no words" of his could pull her from the mirror, suggesting that he tried to get her to help him, but then unselfishly left her to do what she wanted.

Earlier on, while still in the City, Equality/Prometheus shows unselfishness too when he wants to save Liberty/Gaea from the sordid annual procreation ritual. He does this not because he wants her body for himself (though he does love and desire her), but because he wants to spare her the degradation of the ritual.