Lesbians are lesbians because they form loving relationships with women, and gay men are gay because they form loving relationships with men. Neither form relationships because they hate the opposite sex. Lesbians and gay men do not want to be the opposite sex. Within the gay community, there are many people who have challenged and discarded stereotypical gender roles but does not mean they are trying to be the other sex. Being gay, lesbian or bisexual involves celebration and affirmation of one's sex, not a rejection of it.
Some people confuse gayness with transsexuality.
These terms are not mutually exclusive. To understand them, we must understand the difference between biological sex, which is a physical condition at birth, and social gender, which is the body of social expectations about how people ought to dress and behave. Transvestites are people who dress in clothing of the gender role opposite social expectations for their physical sex. Transgenders, transsexuals and transvestites may have any sexual orientation. Studies have shown that people's sexual orientation has no bearing on their mental health and emotional stability.