How to add a boolean datatype column to an existing table in sql?

In SQL SERVER it is BIT, though it allows NULL to be stored

ALTER TABLE person add  [AdminApproved] BIT default 'FALSE';

Also there are other mistakes in your query

1. When you alter a table to add column no need to mention column keyword in alter statement

2. For adding default constraint no need to use SET keyword

3. Default value for a BIT column can be ('TRUE' or '1') / ('FALSE' or 0). TRUE or FALSE needs to mentioned as string not as Identifier