Margaret,

COUNT() in general is slower than EXISTS.

IF EXISTS ( SELECT * from Traveler tr JOIN deleted d on tr.fk_product = d.pk)
	AND EXISTS ( SELECT * from Trans_sernum ts JOIN deleted d on ts.fk_product = d.pk) 
	AND EXISTS ( SELECT * FROM inserted i JOIN DELETED d ON d.pk = i.pk

>There was one modification in the way the pk was being checked. A select had to be used to match the deleted.pk. Below is the actual trigger. .
>
>

>CREATE TRIGGER CK_Product_update ON [Product]
>FOR UPDATE
>AS
>DECLARE @countFk int, @countTrans int
>SELECT @countFk = count(TRAVELER.fk_product)
>        from Traveler where fk_product IN
>	(SELECT PK FROM DELETED)
>
>SELECT @countTrans = count(fk_product)
>        from Trans_sernum where fk_product IN
>	(SELECT PK FROM DELETED)
>
>IF (@countFk > 0 or @countTrans > 0)
>    and EXISTS ( SELECT * FROM inserted i JOIN
>    DELETED d ON d.pk = i.pk
>    WHERE d.fk_engine <> i.fk_engine
>    OR d.fk_assembly <> i.fk_assembly
>    OR d.fk_basematerial <> i.fk_basematerial				
>    OR d.fk_parttype <> i.fk_parttype
>    or d.hpt <> i.hpt
>    or d.stage <> i.stage)
>
>    BEGIN
>        PRINT 'Cannot change Products that have dependents'
>        PRINT 'Transaction has been cancelled'
>        ROLLBACK
>    END
>
>