Disjunctive Syllogism - If one is right, other must be wrong

Do you like disjunctive syllogism?

Of course you do.

Despite complicated name, its actually very simple logic.

It will rain today or I wont bring an umbrella.

If one sentence is true, the other isnt. Very simple.

It is true that it will rain today.

It is false that I wont bring an umbrella.

Homosexuals deserve human rights or homosexuals should be tortured.

It is true that homosexuals deserve human rights, therefore it is false that homosexuals should be tortured.

Disjunctive syllogism is there to show options in arguments. It is usually consisted of 2 opposite options, where one must be true and one must be false.

It is useful in limiting your opponent's case and forcing him to argue for things he doesnt want to argue for.

This logic can be consisted of more than two options, however all options in it must be in the form of "one of these is right, the rest are false" logic.

Presenting options is useful to make case clear and to present the consequences of each option being true or false.

There is no right or wrong way to act that could disqualify one as male or female, queerness is in, ignorance is out, embrace what you are.

Your birth sex if incongruent with the norms you act like combined with feeling out of place and a loathing for one's body mean one must need to transition into the other sex to complete the need the person has to fulfil that gender role.

Incorrect in your example 

If the second sentence is true the first sentence can also be true

Your example:

I am fairly sure that that is on a purely semantical approach, as opposed to pragmatic. How the sentence is structured conveys that, and your deductive reasoning shows the limitations and/or the incorrectness of the sentence.