Meaning : The distinct personality of an individual regarded as a persisting entity.
Example :
You can lose your identity when you join the army.
Synonyms : individuality, personal identity
Meaning : The individual characteristics by which a thing or person is recognized or known.
Example :
Geneticists only recently discovered the identity of the gene that causes it.
It was too dark to determine his identity.
She guessed the identity of his lover.
Meaning : An operator that leaves unchanged the element on which it operates.
Example :
The identity under numerical multiplication is 1.
Synonyms : identity element, identity operator
Meaning : Exact sameness.
Example :
They shared an identity of interests.
Synonyms : identicalness, indistinguishability