Personification gives human-like characteristics to non-human things. The air doesn't have lungs therefore it can't "breathe". The personification here is simply that the air is being described as breathing.
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After getting Cyclops drunk on wine, Odysseus takes away his eye. Cyclops invited Odysseus not to kill him but to bring to his notice that it was Odysseus who was destined to take his eye. Cyclops wanted to befriend him.
