Question about the adab of salam

Assalamu Alaikum
I have a question about the adab of saying “assalamu alaikum” to people of the other gender. I am used to just saying it to anyone who I see that is Muslim, whether I know them or not. Recently I have observed that brothers don’t respond, and someone told me that the brothers should say it first and if they don’t then sisters shouldn’t say it. Is this true? The same brothers that don't respond to salam, however, don’t have a problem talking to non-muslim girls, and they are always talking to muslim sisters online or in email. I don’t understand why , then, they would have a problem saying assalamu alaikum. Isn’t it our haq upon each other to give salam?
I guess I went a little off topic, but my question is what is the adab of giving salams to people of the other gender?

[-ilmseeker-]

Assalamu Alaikum,

I had a similar question as well. A brother once told me that it was stated in a hadith that muslim men should only say a simple 'salam' to a sister instead of the entire Islamic greeting - Assalamu Alaikum. He never provided me with the hadith, however I was wondering if this was infact true?
Also, can one say 'salam' to a non-muslim? I was always under the impression that we can, however I was just told recently that we weren't suppose to.