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Understanding the Union of Sets When One is a Subset
Understanding the Union of Sets When One is a Subset
Understanding the relationship between the union of sets when one set is a subset of another is a fundamental concept in set theory, which is crucial for both theoretical explorations and practical applications in various fields such as computer science, data analysis, and even everyday mathematical reasoning. In this article, we will explore the conditions under which the union of a subset and its larger set equals the larger set, and provide a detailed proof to solidify our understanding.
Definitions
Before diving into the proof, let's define the terms we will be working with:
Subset: A set (A) is a subset of set (B), denoted as (A subseteq B), if every element of set (A) is also an element of set (B). Union of Sets: The union of two sets (A) and (B), written as (A cup B), is the set containing all elements from both (A) and (B).Proving (A subseteq B implies A cup B B)
Let's first consider the case where (A emptyset), the empty set. If (A emptyset), then the union of (A) with any set (B) must be simply (B) since there are no elements in (A) to add to (B).
Case 1: (A emptyset)
Assume that A {} (the empty set). Then it is clear that A U B B because a union with an empty set adds nothing.
For the more general case, let's prove this by contradiction. Assume that (A cup B) contains at least one element (x) that is not in (B). Since (x) is not in (B) and (A) is a subset of (B), (x) is also not in (A). However, by definition of the union, (x) must be in (A cup B). This is a contradiction, hence (A cup B B).
Case 2: (A subseteq B)
Now, let's consider the case where (A) is a proper subset of (B). To prove that (A cup B B), we will use logical connectives and definitions.
Apply the definitions of union and subset.Obviously, B is in A U B.If A is a subset of B, then x in A implies x in B.x in A U B implies x in A or x in B, which implies x in B or x in B, which is x in B.
Step-by-step proof:
Step 1: By definition, the union (A cup B) contains all elements that are in either (A) or (B) (or both). Step 2: Since (A subseteq B), every element in (A) is also in (B). Therefore, any element that is in (A cup B) must either be in (A) or in (B). Step 3: But since (A subseteq B), any element that is in (A) is also in (B), so (A) does not add any new elements to (B) in the union. Conclusion: Therefore, (A cup B) contains exactly the same elements as (B), which means (A cup B B).Example Illustration
Let's provide a concrete example to illustrate the proof:
Example:
Let A {1, 2, 3} and B {1, 2, 3, 4, 5}.Here, A is a subset of B. A U B {1, 2, 3, 4, 5} BHence, it is proven that A U B B.
Conclusion
In conclusion, if set (A) is a subset of set (B), then the union of (A) and (B) is equal to (B). This property is crucial for understanding the behavior of sets in various mathematical and practical contexts. Whether you are working with data, programming logic, or theoretical mathematics, knowing when and how to apply this property can simplify many problems.