The __________ contains all the objects or elements under consideration and the empty set, sometimes called the null set, is the set that contains no elements

The term that fits in the blank is "universe" or "universal set."

In set theory, the universal set is the set that contains all possible objects or elements relevant to a particular discussion or problem. It serves as the overarching set from which subsets can be drawn. For instance, if we are discussing a set of all natural numbers, the universal set would include every natural number that one could consider, and potentially other number types depending on context (like integers, rational numbers, etc.).

Furthermore, the empty set, denoted as ∅ or { }, is a special set that contains no elements. It is a fundamental concept in mathematics and is considered a subset of every set, including the universal set. The implications of having an empty set are significant in various branches of mathematics, including logic, statistics, and computer science, as it often represents the idea of "nothingness" or "absence" within set theory.

Understanding these two concepts is crucial, as they underpin much of the language and operations involved in working with sets, such as unions, intersections, and differences.