A member of the family dying is a situation I would not like

dying 做主语补足语,修饰 a member of the family 。不充当句子主谓宾成分。is为谓语,situation为宾语,也是定语从句的先行词,其后为一个定语从句,situation后省略了which或that。至于dying这种用法,eg。The boy playing football on the playground is my classmate。playing football on the playground 即是修饰the boy 的。