Login based on different browsers

If the users logins on chrome and then switches to firefox and opens my website should I ask him to login again? I think in facebook, there are different sessions maintained for each browser. Or when the user opens facebook on firefox he is asked to login again and if he logs out from facebook on firefox, he remains signed on chrome.

Is this the right solution ?

Browsers don’t share cookies, so if a user switches to another browser they will have to log in again.