Re: Assalam-o-Alaikum
Ibrahim (as) said the 'complete' salam to his father/uncle despite the later being a polytheist.
قَالَ أَرَاغِبٌ أَنْتَ عَنْ آلِهَتِي يَا إِبْرَاهِيمُ ۖ لَئِنْ لَمْ تَنْتَهِ لَأَرْجُمَنَّكَ ۖ وَاهْجُرْنِي مَلِيًّا
[19:46] He said: Do you dislike my gods, O Ibrahim? If you do not desist I will certainly revile you, and leave me for a time.
قَالَ** سَلَامٌ عَلَيْكَ ۖ** سَأَسْتَغْفِرُ لَكَ رَبِّي ۖ إِنَّهُ كَانَ بِي حَفِيًّا
[19:47] He said: Peace be on you, I will pray to my Lord to forgive you; surely He is ever Affectionate to me:
Here you go. Assalam-o-Alaikum is nothing but a prayer for the addressee. As psyah said
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When I say it to a Muslim i say it with the intention that "Allah (SWT) increases their state of peace" and when I say it to a non-Muslim, I say it with the intention that "Allah (SWT) brings them to a state of peace" ...
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same is the case with AoA. you are not giving away anything from your side. It is just a prayer and it is upto Allah what to do about it.
Another verse:
**4:87) "When you are greeted with a greeting, greet in return with what is better than it, or (at least) return it equally”
**
I think there are two instances I can think of where above is not applicable
1) The person before you greet you in insulting way or as Jews at the time of prophet (pbuh) used to say. say 'Assam-o-Alaikum'. In that case, keep quiet or say wa-alaikum.
2) The person greet you with salam but is known to be insulting and abusive towards Islam, Allah(swt), prophet (pbuh) etc.
Once the promised messiah(as) was doing his ablution before prayer. a hindu pandit lekhraam was passing by. He said salam to promised messiah (as). He didn't respond. He repeated salam a few times but hazrat sahib didn't reply. He went away. one of the companions said. Why didn't you reply to his salam? The promised Messiah(as) said: "Hamaare aaqa(pbuh) ko to gaaliaan deta hai aur hamein salam karta hai".