kɛː/Submit
noun
1.
the provision of what is necessary for the health, welfare, maintenance, and protection of someone or something.
"the care of the elderly"
synonyms:safe keeping, supervision, custody, charge, protection, keeping, keep, control, management, ministration, guidance, superintendence, tutelage, aegis, responsibility; More
2.
serious attention or consideration applied to doing something correctly or to avoid damage or risk.
"he planned his departure with great care"
synonyms:caution, carefulness, wariness, awareness, heedfulness, heed, attention, attentiveness, alertness, watchfulness, vigilance, circumspection, prudence, guardedness, observance More
verb
1.
feel concern or interest; attach importance to something.
"they don't care about human life"
synonyms:be concerned, worry (oneself), trouble oneself, bother, mind; More
2.
look after and provide for the needs of.
"he has numerous animals to care for"
synonyms:look after, take care of, tend, attend to, mind, minister to, take charge of, nurse, provide for, foster, protect, watch, guard; be responsible for, keep safe, keep an eye on; sit with, babysit, childmind
"a hospice which cares for the terminally ill"
So what does it mean when we say we don't care about someone?
I guess it means that their actions no longer affect us in anyways. Personally I think it's an emotion or a (lack thereof - pun fully intended) feeling that signifies you pretty much gave up on that person. It's sad. Even hating someone brings out something deep down, but when we no longer care, we just don't.
