Let A = {2, 4, 6, 8, 10, 12}, B = {2, 4, 8, 10}, C = {4, 10, 12}, D = {2, 10}, andU = {2, 4, 6, 8, 10, 12, 14}. Determine whether each statement is true or false. ∅ ⊆ ∅

Verified step by step guidance

1. Recognize that the symbol '∅' represents the empty set, which contains no elements.
2. Understand the subset relation '⊆', which indicates that all elements of the first set are also elements of the second set.
3. Consider that the empty set, being devoid of elements, does not contain any element that could potentially not be in another set, including another empty set.
4. Apply the subset definition to the empty set: Since there are no elements in ∅ that could fail to be in another ∅, ∅ is a subset of ∅.
5. Conclude that the statement ∅ ⊆ ∅ is true because the empty set is always considered a subset of itself.

Key Concepts

Set Theory

Set theory is a fundamental area of mathematics that deals with the study of sets, which are collections of objects. It provides the basic language and operations for understanding relationships between different groups of elements, including concepts like subsets, unions, intersections, and complements. Understanding set theory is essential for analyzing statements about sets, such as whether one set is a subset of another.

Subset Definition

A subset is defined as a set where every element of the subset is also an element of another set. The notation 'A ⊆ B' indicates that set A is a subset of set B. Importantly, the empty set (∅) is considered a subset of every set, including itself, which is a crucial aspect of set theory that helps in evaluating statements about subsets.

Empty Set

The empty set, denoted as ∅, is a unique set that contains no elements. It plays a significant role in set theory as it is the identity element for the union operation and a subset of every set. Understanding the properties of the empty set is vital for determining the truth of statements involving subsets, particularly in the context of the question provided.