Given up refers to the act of relinquishing something, whether it be an object or a concept. There are various synonyms for this phrase that can be used to express similar ideas. One common phrase is "surrendered," which implies that the person has yielded control or ownership. Another option is "abandoned," which connotes a sense of desertion. "Renounced" also works, as it suggests the person has turned their back on something they once held dear. Lastly, "forfeited" is a synonym that implies the loss of something valuable or important. Each of these synonyms for given up can be used to add variety to one's writing or speech.