Intersection Subset of Union

Prove that the intersection of $A$ and $B$ is a subset of $A$ union $B$.

In set theory, understanding the relationships between different operations on sets is fundamental. This problem addresses the concept of proving subset relationships, specifically focusing on the intersection and union of sets. To approach this problem, you need a solid grasp of what it means for one set to be a subset of another. A set X is considered a subset of set Y if every element of X is also an element of Y. This forms the basis of the argument you will provide in your proof.

The key to this exercise is to delve into the definitions of intersection and union. The intersection of two sets A and B, denoted as A intersection B, contains all elements that are common to both A and B. On the other hand, the union of A and B, represented as A union B, includes all elements that are in either A, B, or both. Once these concepts are clear, the task becomes proving that any element found in A intersection B must also be found in A union B, leading to the conclusion that the intersection is indeed a subset of the union.

This type of proof typically involves logical reasoning and clear articulation of each step, emphasizing the foundational role of definitions in mathematical proofs. Understanding these underlying principles is crucial as they form the building blocks for more complex operations and theorems in set theory. By mastering these basic relations, students can better tackle more advanced topics in discrete mathematics.