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Chapter 8: Problem 46

Find each indicated intersection or union. $$ \\{3,5,7\\} \cap \varnothing $$

Short Answer

Expert verified
The intersection \( \{3, 5, 7\} \cap \varnothing \) is \( \varnothing \).

Step by step solution

01

Understand the Symbols

The symbol \(\cap\) represents the intersection of two sets. The intersection is a new set containing all elements that are common to both sets.
02

Identify the Sets

In this case, the first set is \(\{3, 5, 7\}\), and the second set is the empty set, denoted by \(\varnothing\).
03

Find Common Elements

To find the intersection \( \{3, 5, 7\} \cap \varnothing \), look for elements that are present in both sets. The empty set \( \varnothing \) has no elements, so there can be no common elements between \( \{3, 5, 7\} \) and \( \varnothing \).
04

Conclusion

Since there are no common elements, the intersection of \( \{3, 5, 7\} \) and \( \varnothing \) is also the empty set, denoted by \( \varnothing \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Intersection of Sets
An intersection of sets refers to the set of elements that are common in both sets. Let's take two sets, A and B. The intersection of A and B is written as \(A \cap B\). The resulting set will only include elements that are in both A and B. For example, if \(A = \{1, 2, 3\}\) and \(B = \{2, 3, 4\}\), the intersection \(A \cap B\) will be \(\{2, 3\}\). This concept is fundamental in set theory and helps us understand shared characteristics between different sets.
In our specific exercise, we are asked to find the intersection of the set \( \{3, 5, 7\} \) and the empty set \( \varnothing \). By understanding the properties of the empty set, we can clearly see the outcome.
Empty Set
The empty set, denoted by \( \varnothing \), is a unique set with no elements within it. It鈥檚 like an empty container; no matter how much you look inside, you won't find anything. Understanding the empty set is crucial when working with intersections because the empty set acts as a neutral element. Let's go back to our exercise where we look for the intersection between \( \{3, 5, 7\} \) and \( \varnothing \). Since \( \varnothing \) contains no elements, it means that there are no common elements between any set and the empty set. As such, the intersection of any set with \( \varnothing \) will always be \( \varnothing \) itself.
Common Elements
Common elements are the foundation of intersecting sets. When finding the intersection, we focus on identifying which elements appear in all involved sets. If no elements are found in all sets, the intersection will be the empty set. For instance, in the exercise \( \{3, 5, 7\} \cap \varnothing \), there are no elements in the empty set \( \varnothing \). Thus, it鈥檚 impossible to find any common elements between \( \{3, 5, 7\} \) and \( \varnothing \). This confirms that the intersection is also \( \varnothing \). This process emphasizes the importance of common elements in determining intersections and helps us better understand how different sets relate to one another.

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