As a Muslim who live among non-Muslim (majority) both at home and at work, how can a Muslim relate with a non-Muslim? Considering the fact that their tradition is not compatible with that of Islaam. For example, shaking hand with especially opposite sex, embarrassing inform of greeting, eating together, etc.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

If the questioner means that he lives in a non-Muslim environment at home and at work, i.e. in a non-Muslim country, then the disadvantages that result in mixing with non-Muslims are various and one cannot get rid of these disadvantages except by leaving these people and migrating from their land. That's why the Prophet warned us against living amongst them as he said: "Whoever mixes with a non-Muslim and resides with him, he is like him." [Abu Daawood]

The Prophet also said: "I am free from every Muslim who stays among the polytheist." [Abu Daawood] So it is an obligation for whoever lives in a non-Muslim land and worries about his religion to migrate to Muslim countries. But if he is in a Muslim country where many acts of disobedience and prohibitions occur, then it is an obligation on him to stay steadfast on the Sharee'ah of Allaah and not participate in falsehood and prohibitions. He must also enjoin good and prohibit evil according to his ability. He has to demonstrate to his friends, colleagues and family members that such traditions like shaking hands with marriageable women, women not wearing Hijab and mixing, etc… all of this is forbidden in Islam. So if they act according to this, it is best, otherwise it is safer for your religion not to mix with them. But you have to give a Muslim his rights, like visiting him if he is sick, and saying 'may Allaah have mercy on you' if he sneezes (after thanking Allaah), follow his funeral procession, and so on. But it is better not to mix with them. Allaah says to his Prophet (interpretation of meaning) {Show forgiveness, enjoin what is good, and turn away from the foolish (i.e. don't punish them).}[7:199].

Allaah knows best.