Problem 10 - Algebra for College Students

Question

Let \(A=\\{1,2,3,4,5,6\\}, B=\\{1,3,5\\}, C=\\{1,6\\},\) and \(D=\\{4\\} .\) Find each set. $$ A \cap \varnothing $$

Step-by-Step Solution

Answer

The intersection of set \( A \) and \( \varnothing \) is \( \varnothing \).

Step 1 - Understand the Given Sets

The problem provides four sets: \(A = \{1, 2, 3, 4, 5, 6\}\), \(B = \{1, 3, 5\}\), \(C = \{1, 6\}\), and \(D = \{4\}\). Additionally, it includes the empty set, \( \varnothing \). In this task, the intersection of set \( A \) with the empty set \( \varnothing \) is required.

Step 2 - Recall the Property of Intersection with the Empty Set

Any set \( X \) intersected with the empty set \( \varnothing \) results in the empty set: \( X \cap \varnothing = \varnothing \). This is because \( \varnothing \) contains no elements, so there are no elements that can also be in \( X \).

Step 3 - Compute the Intersection

Using the property from Step 2, intersect set \( A \) with the empty set: \( A \cap \varnothing = \varnothing \). Therefore, the intersection of \( A \) and \( \varnothing \) is \( \varnothing \).

Key Concepts

Empty Set

An empty set, denoted as \( \varnothing \) or \( \{\} \), is a set that contains no elements. This is a fundamental concept in set theory and is crucial in understanding various operations involving sets. For instance, when a set is intersected with the empty set, no matter what elements the initial set contains, the result of the intersection will always be the empty set.
This is because there are no elements in the empty set to share with any other set to form an intersection.
For example:

• \( \{1, 2, 3\} \cap \varnothing = \varnothing \)
• \( \{a, b, c\} \cap \varnothing = \varnothing \)

Thus, it's essential to recognize that the empty set signifies the absence of elements.

Intersection of Sets

In algebra and set theory, the intersection of sets is an operation that combines two sets to form a new set containing only the elements that are present in both original sets. It's denoted using the symbol \( \cap \).
For example, if you have two sets, \( A = \{1, 2, 3\} \) and \( B = \{2, 3, 4\} \), then the intersection \( A \cap B = \{2, 3\} \) because 2 and 3 are the common elements present in both sets.

• Notation: The intersection is written as \( A \cap B \)
• Result: The new set formed contains only common elements

It's important to remember that if two sets have no elements in common, their intersection will be the empty set, denoted as \( \varnothing \).
For example, \( \{1, 2\} \cap \{3, 4\} = \varnothing \).

Properties of Set Operations

Set operations have several key properties that are useful for solving problems in algebra. Understanding these properties can simplify the process of working with sets.
Here are some of the main properties:

• Commutative Property: For intersection and union, the order of sets doesn't matter. \( A \cap B = B \cap A \) and \( A \cup B = B \cup A \).
• Associative Property: Groups of sets can be grouped in any way. \( (A \cap B) \cap C = A \cap (B \cap C) \) and \( (A \cup B) \cup C = A \cup (B \cup C) \).
• Identity Property: The intersection of any set with itself yields the same set. \( A \cap A = A \). Similarly, the union of any set with itself also yields the same set. \( A \cup A = A \).
• Distributive Property: Intersection distributes over union and vice versa. \( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \) and \( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \).
• Null Property: The intersection of any set with the empty set is the empty set. \( A \cap \varnothing = \varnothing \).

These properties help streamline computations and provide a consistent framework for solving problems involving sets.