How to save a multiple checkbox array into a non-hierarchical taxonomy

by

First in checkbox value use taxonomy id not the name of taxonomy.

For save more than one taxonomy relation related to a post in wp_term_relationship table use

$theme = $_POST['specialities'];
wp_set_object_terms( $post_id, $theme, 'activity_category' );

“wp_set_object_terms” will save checked taxonomy value to “wp_term_relationship” table with post id.