trigger certificationToFailedNew on Exam__c (after update) {
    //List out all the exams associated with the parent of the triggered object. exam record -> ParentID -> other child records which share the same parent
    List<Exam__c> exams = new List<Exam__c>();
    exams = [
        SELECT Id,Exam_id__c, Exam_state__c, Certification__r.Id, Certification__r.Exam_state__c
        FROM Exam__c
        WHERE Exam_state__c in ('Failed') AND Certification__r.Exam_state__c in ('Active') AND Certification__r.id == Certification__c.id IN :trigger.newMap.keySet()
            AND id != id IN :trigger.newMap.keySet()];

trigger certificationToFailedNew on Exam__c (after update) {

    set<id> certificationId = new set<id>();
    String x;
    String y;

    for(Exam__c e : Trigger.new){
        certificationId.add(e.Certification__c);
        x = e.Exam_id__c;
        y = e.Exam_state__c;
    }

    List<Exam__c> exams = new List<Exam__c>();
    exams = [
        SELECT Id, Exam_id__c, Exam_state__c, Certification__r.Id, Certification__r.Exam_state__c
        FROM Exam__c
        WHERE Exam_state__c in ('Failed') AND Certification__r.Exam_state__c in ('Active')
            AND Id NOT IN :trigger.newMap.keySet() AND Certification__r.Id IN :certificationId
            AND Exam_id__c = :x AND Exam_state__c = :y];
}

WHERE Exam_state__c in ('Failed') AND Certification__r.Exam_state__c in ('Active')
        AND Certification__r.id == Certification__c.id IN :trigger.newMap.keySet()
        AND id != id IN :trigger.newMap.keySet()];

Certification__c NOT IN :setOfParents

Certification__c IN :setOfParents AND id NOT IN :trigger.newMap.keySet()

trigger certificationToFailedNew on Exam__c (after update) {
    set<id> certificationId = new set<id>();
    for(Exam__c e : Trigger.new){
        certificationId.add(e.Certification__c);
    }
    List<Exam__c> exams = [SELECT Id,Exam_id__c, Exam_state__c,Certification__r.Id,
                           Certification__r.Exam_state__c FROM Exam__c
                           WHERE Exam_state__c = 'Failed' AND Certification__r.Exam_state__c = 'Active'
                           AND Certification__r.id  IN :certificationId AND id NOT IN :trigger.newMap.keySet()];
}